Willmore Spheres in Compact Riemannian Manifolds
Mondino, Andrea
2012-01-01
The paper is devoted to the variational analysis of the Willmore, and other L^2 curvature functionals, among immersions of 2-dimensional surfaces into a compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is twofold, on one hand, we give the right setting for doing the calculus of variations (including min max methods) of such functionals for immersions into manifolds and, on the other hand, we prove existence results for possibly branched Willmore spheres under various constraints (prescribed homotopy class, prescribed area) or under curvature assumptions for M^m. To this aim, using the integrability by compensation, we develop first the regularity theory for the critical points of such functionals. We then prove a rigidity theorem concerning the relation between CMC and Willmore spheres. Then we prove that, for every non null 2-homotopy class, there exists a representative given by a Lipschitz map from the 2-sphere into M^m realizing a connected family of conformal smooth (possibly branche...
Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds
Institute of Scientific and Technical Information of China (English)
LI Yuxiang
2005-01-01
Let (M, g) be a compact Riemannian manifold without boundary, and (N, g) a compact Riemannian manifold with boundary. We will prove in this paper that the ∫MudVg=o,sup∫M|(△↓)u|ndVg=1∫Meαn|u|n/n-1dVg,∫M(|(△↓)|n+|u|n)dVg=1∫Meαn|u|n/n-1dVg,and u|(e)N=0,∫Msup|(△↓)u|dVgN=1∫Neαn|u|n/n-1dVgn can be attained. Our proof uses the blow-up analysis.
Lin, Tong; Zha, Hongbin
2008-05-01
Recently, manifold learning has been widely exploited in pattern recognition, data analysis, and machine learning. This paper presents a novel framework, called Riemannian manifold learning (RML), based on the assumption that the input high-dimensional data lie on an intrinsically low-dimensional Riemannian manifold. The main idea is to formulate the dimensionality reduction problem as a classical problem in Riemannian geometry, i.e., how to construct coordinate charts for a given Riemannian manifold? We implement the Riemannian normal coordinate chart, which has been the most widely used in Riemannian geometry, for a set of unorganized data points. First, two input parameters (the neighborhood size k and the intrinsic dimension d) are estimated based on an efficient simplicial reconstruction of the underlying manifold. Then, the normal coordinates are computed to map the input high-dimensional data into a low-dimensional space. Experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data, preserve radial geodesic distances, and yield regular embeddings.
Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
[1]Trudinger, N. S., On embedding into Orlicz space and some applications, J. Math. Mech., 1967, 17: 473-484.[2]Moser, J., A sharp form of an Inequality by N.Trudinger, Ind. Univ. Math. J., 1971, 20: 1077-1091.[3]Adams, D. R., A sharp inequality of J. Moser for higher order derivatives, Anna. Math., 1988, 128: 385-398.[4]Fontana, L., Sharp borderline Sobolev inequalities on compact Riemannian manifolds. Comm. Math. Helv.,1993, 68: 415- 454.[5]Lin, K.C., Extremal functions for Moser's inequality, Trans. Amer.Math. Sco., 1996, 348: 2663-2671.[6]Carleson, L., Chang, S. Y. A., On the existence of an extremal function for an inequality of J.Moser, Bull. Sc.Math., 1986, 110: 113-127.[7]Flucher, M., Extremal functions for Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv., 1992,67: 471-497.[8]Adimurth, Struwe, M., Global compactness properties of semilinear elliptic equations with critical exponential growth, J. Funct. Anal., 2000, 175(1): 125-167.[9]Li,Y., Moser-Trudinger inequality on manifold of dimesion two, J. Partial Differential Equations, 2001, 14(2):163-192.[10]Kichenassamy, S., Veron, L., Singular solutions of the p-laplace equation, Math. Ann., 1986, 275: 599-615.[11]Ding, W. Y., Jost, J., Li, J. et al, The differential equation -△u = 8π - 8πheu on a compact Riemann Surface,Asia. J. Math., 1997, 1(2): 230-248.[12]Tolksdorf, P., Regularity for a more general class of qusilinear elliptic equations, J. D. E., 1984, 51:126-150.[13]Serrin, J., Local behavior of solutions of qusai-linear equations, Acta. Math., 1964, 111: 248-302.[14]Struwe, M., Positive solution of critical semilinear elliptic equations on non-contractible planar domain, J. Eur.Math. Soc., 2000, 2(4): 329-388.[15]Serrin, J., Isoled singularities of solutions of quasilinear equations, Acta. Math., 1965, 113: 219-240.[16]Struwe, M., Critical points of embedding of H01,n into Orlicz space, Ann. Inst. Henri., 1988, 5(5): 425-464.[17]Chen, W. X., Li, C., Classification of solutions of
Instability of elliptic equations on compact Riemannian manifolds with non-negative Ricci curvature
Directory of Open Access Journals (Sweden)
Arnaldo S. Nascimento
2010-05-01
Full Text Available We prove the nonexistence of nonconstant local minimizers for a class of functionals, which typically appear in scalar two-phase field models, over smooth N-dimensional Riemannian manifolds without boundary and non-negative Ricci curvature. Conversely, for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative, we prove the existence of nonconstant local minimizers for the same class of functionals.
Principal Curves on Riemannian Manifolds
DEFF Research Database (Denmark)
Hauberg, Søren
2015-01-01
Euclidean statistics are often generalized to Riemannian manifolds by replacing straight-line interpolations with geodesic ones. While these Riemannian models are familiar-looking, they are restricted by the inflexibility of geodesics, and they rely on constructions which are optimal only...... in Euclidean domains. We consider extensions of Principal Component Analysis (PCA) to Riemannian manifolds. Classic Riemannian approaches seek a geodesic curve passing through the mean that optimize a criteria of interest. The requirements that the solution both is geodesic and must pass through the mean tend...... from Hastie & Stuetzle to data residing on a complete Riemannian manifold. We show that for elliptical distributions in the tangent of spaces of constant curvature, the standard principal geodesic is a principal curve. The proposed model is simple to compute and avoids many of the pitfalls...
Principal Curves on Riemannian Manifolds.
Hauberg, Soren
2016-09-01
Euclidean statistics are often generalized to Riemannian manifolds by replacing straight-line interpolations with geodesic ones. While these Riemannian models are familiar-looking, they are restricted by the inflexibility of geodesics, and they rely on constructions which are optimal only in Euclidean domains. We consider extensions of Principal Component Analysis (PCA) to Riemannian manifolds. Classic Riemannian approaches seek a geodesic curve passing through the mean that optimizes a criteria of interest. The requirements that the solution both is geodesic and must pass through the mean tend to imply that the methods only work well when the manifold is mostly flat within the support of the generating distribution. We argue that instead of generalizing linear Euclidean models, it is more fruitful to generalize non-linear Euclidean models. Specifically, we extend the classic Principal Curves from Hastie & Stuetzle to data residing on a complete Riemannian manifold. We show that for elliptical distributions in the tangent of spaces of constant curvature, the standard principal geodesic is a principal curve. The proposed model is simple to compute and avoids many of the pitfalls of traditional geodesic approaches. We empirically demonstrate the effectiveness of the Riemannian principal curves on several manifolds and datasets.
The convexity radius of a Riemannian manifold
Dibble, James
2014-01-01
The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characterization of the convexity radius that resembles a classical result of Klingenberg about the injectivity radius.
Minimal Webs in Riemannian Manifolds
DEFF Research Database (Denmark)
Markvorsen, Steen
2008-01-01
are of instrumental importance for the applications. We apply these properties to show that minimal webs in ambient Riemannian spaces share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in such spaces. In particular we use appropriate versions of the divergence......)$ into Riemannian manifolds $(N^{n}, h)$. Such immersions we call {\\em{minimal webs}}. They admit a natural 'geometric' extension of the intrinsic combinatorial discrete Laplacian. The geometric Laplacian on minimal webs enjoys standard properties such as the maximum principle and the divergence theorems, which...... theorems together with the comparison techniques for distance functions in Riemannian geometry and obtain bounds for the first Dirichlet eigenvalues, the exit times and the capacities as well as isoperimetric type inequalities for so-called extrinsic $R-$webs of minimal webs in ambient Riemannian manifolds...
Polynomial Regression on Riemannian Manifolds
Hinkle, Jacob; Fletcher, P Thomas; Joshi, Sarang
2012-01-01
In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Bookstein as well as the analysis of the shape changes associated with aging of the corpus callosum from the OASIS Alzheimer's study.
Natural Connections on Riemannian Product Manifolds
Gribacheva, Dobrinka
2011-01-01
A Riemannian almost product manifold with integrable almost product structure is called a Riemannian product manifold. In the present paper the natural connections on such manifolds are studied, i.e. the linear connections preserving the almost product structure and the Riemannian metric.
Roughly isometric minimal immersions into Riemannian manifolds
DEFF Research Database (Denmark)
Markvorsen, Steen
A given metric (length-) space $X$ (whether compact or not) is roughly isometric to any one of its Kanai graphs $G$, which in turn can be {\\em{geometrized}} by considering each edge of $G$ as a 1-dimensional manifold with an associated metric $g$ giving the 'correct' length of the edge. In this t......A given metric (length-) space $X$ (whether compact or not) is roughly isometric to any one of its Kanai graphs $G$, which in turn can be {\\em{geometrized}} by considering each edge of $G$ as a 1-dimensional manifold with an associated metric $g$ giving the 'correct' length of the edge....... In this talk we will mainly be concerned with {\\em{minimal}} isometric immersions of such geometrized approximations $(G, g)$ of $X$ into Riemannian manifolds $N$ with bounded curvature. When such an immersion exists, we will call it an $X$-web in $N$. Such webs admit a natural 'geometric' extension...
Some properties of Fr\\'echet medians in Riemannian manifolds
Yang, Le
2011-01-01
The consistency of Fr\\'echet medians is proved for probability measures in proper metric spaces. In the context of Riemannian manifolds, assuming that the probability measure has more than a half mass lying in a convex ball and verifies some concentration conditions, the positions of its Fr\\'echet medians are estimated. It is also shown that, in compact Riemannian manifolds, the Fr\\'echet sample medians of generic data points are always unique.
MOSER-TR DINGER INEQUALITY ON COMPACT RIEMANNIAN MANIFOLDS OF DIMENSION TWO
Institute of Scientific and Technical Information of China (English)
Li Yuxiang
2001-01-01
In this paper, we prove Moser-Triidinger inequality in any two dimen- sional manifolds. Let (M,gM) be a two dimensional manifold without boundary and (g, gN) with boundary, we shall prove the following three inequalities: Moreover, we shall show that there exist of extremal functions which attain the above three inequalities.
Characterizing humans on Riemannian manifolds.
Tosato, Diego; Spera, Mauro; Cristani, Marco; Murino, Vittorio
2013-08-01
In surveillance applications, head and body orientation of people is of primary importance for assessing many behavioral traits. Unfortunately, in this context people are often encoded by a few, noisy pixels so that their characterization is difficult. We face this issue, proposing a computational framework which is based on an expressive descriptor, the covariance of features. Covariances have been employed for pedestrian detection purposes, actually a binary classification problem on Riemannian manifolds. In this paper, we show how to extend to the multiclassification case, presenting a novel descriptor, named weighted array of covariances, especially suited for dealing with tiny image representations. The extension requires a novel differential geometry approach in which covariances are projected on a unique tangent space where standard machine learning techniques can be applied. In particular, we adopt the Campbell-Baker-Hausdorff expansion as a means to approximate on the tangent space the genuine (geodesic) distances on the manifold in a very efficient way. We test our methodology on multiple benchmark datasets, and also propose new testing sets, getting convincing results in all the cases.
Cohomogeneity Two Actions on Flat Riemannian Manifolds
Institute of Scientific and Technical Information of China (English)
R. MIRZAIE
2007-01-01
In this paper, we study fiat Riemannian manifolds which have codimension two orbits,under the action of a closed and connected Lie group G of isometries. We assume that G has fixedpoints, then characterize M and orbits of M.
Radio Interferometric Calibration Using a Riemannian Manifold
Yatawatta, Sarod
2013-01-01
In order to cope with the increased data volumes generated by modern radio interferometers such as LOFAR (Low Frequency Array) or SKA (Square Kilometre Array), fast and efficient calibration algorithms are essential. Traditional radio interferometric calibration is performed using nonlinear optimization techniques such as the Levenberg-Marquardt algorithm in Euclidean space. In this paper, we reformulate radio interferometric calibration as a nonlinear optimization problem on a Riemannian manifold. The reformulated calibration problem is solved using the Riemannian trust-region method. We show that calibration on a Riemannian manifold has faster convergence with reduced computational cost compared to conventional calibration in Euclidean space.
Harmonic Riemannian Maps on Locally Conformal Kaehler Manifolds
Indian Academy of Sciences (India)
Bayram Sahin
2008-11-01
We study harmonic Riemannian maps on locally conformal Kaehler manifolds ($lcK$ manifolds). We show that if a Riemannian holomorphic map between $lcK$ manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the $lcK$ manifold is Kaehler. Then we find similar results for Riemannian maps between $lcK$ manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.
Canonical connection on a class of Riemannian almost product manifolds
Mekerov, Dimitar
2009-01-01
The canonical connection on a Riemannian almost product manifolds is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Riemannian almost product manifolds with nonintegrable almost product structure.
Poincare duality angles for Riemannian manifolds with boundary
Shonkwiler, Clayton
2009-01-01
On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and boundary of the manifold. The principal angles between these interior subspaces are all acute and are called Poincare duality angles. This paper determines the Poincare duality angles of a collection of interesting manifolds with boundary derived from complex projective spaces and from Grassmannians, providing evidence that the Poincare duality angles measure, in some sense, how "close" a manifold is to being closed. This paper also elucidates a connection between the Poincare duality angles and the Dirichlet-to-Neumann operator for differential forms, which generalizes the classical Dirichlet-to-Neumann map arising in the problem of Electrical Impedance Tomography. Specifically, the Poincare duality...
Convex functions and optimization methods on Riemannian manifolds
Udrişte, Constantin
1994-01-01
This unique monograph discusses the interaction between Riemannian geometry, convex programming, numerical analysis, dynamical systems and mathematical modelling. The book is the first account of the development of this subject as it emerged at the beginning of the 'seventies. A unified theory of convexity of functions, dynamical systems and optimization methods on Riemannian manifolds is also presented. Topics covered include geodesics and completeness of Riemannian manifolds, variations of the p-energy of a curve and Jacobi fields, convex programs on Riemannian manifolds, geometrical constructions of convex functions, flows and energies, applications of convexity, descent algorithms on Riemannian manifolds, TC and TP programs for calculations and plots, all allowing the user to explore and experiment interactively with real life problems in the language of Riemannian geometry. An appendix is devoted to convexity and completeness in Finsler manifolds. For students and researchers in such diverse fields as pu...
Natural connections on conformal Riemannian P-manifolds
Gribacheva, Dobrinka
2011-01-01
The class of conformal Riemannian P-manifolds is the largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric. This class is an analogue of the class of conformal Kaehler manifolds in almost Hermitian geometry. In the present work we study on a conformal Riemannian P-manifold (M, P, g) the natural linear connections, i.e. the linear connections preserving the almost product structure P and the Riemannian metric g. We find necessary and sufficient conditions the curvature tensor of such a connection to have similar properties like the ones of the Kaehler tensor in Hermitian geometry. We determine the type of the manifolds admitting a natural connection with a parallel torsion.
General formula for lower bound of the first eigenvalue on Riemannian manifolds
Institute of Scientific and Technical Information of China (English)
陈木法; 王凤雨
1997-01-01
A general formula for the lower bound of the first eigenvalue on compact Riemannian manifolds is presented. The formula improves the main known sharp estimates including Lichnerowicz’s estimate and Zhong-Yang’s estimate. Moreover, the results are extended to the noncompact manifolds. The study is based on the probabilistic approach (i.e. the coupling method).
P-connection on Riemannian almost product manifolds
Mekerov, Dimitar
2009-01-01
In the present work, we introduce a linear connection (preserving the almost product structure and the Riemannian metric) on Riemannian almost product manifolds. This connection, called P-connection, is an analogue of the first canonical connection of Lichnerowicz in the Hermitian geometry and the B-connection in the geometry of the almost complex manifolds with Norden metric. Particularly, we consider the P-connection on a the class of manifolds with nonintegrable almost product structure.
Stochastic gradient descent on Riemannian manifolds
Bonnabel, Silvere
2011-01-01
Stochastic gradient descent is a simple appproach to find the local minima of a function whose evaluations are corrupted by noise. In this paper, mostly motivated by machine learning applications, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and we show several well-known algorithms can be cast in our versatile geometric framework. We also address the gain tuning issue in connection with the tools of the recent theory of symmetry-preserving observers.
The Identification of Convex Function on Riemannian Manifold
Directory of Open Access Journals (Sweden)
Li Zou
2014-01-01
Full Text Available The necessary and sufficient condition of convex function is significant in nonlinear convex programming. This paper presents the identification of convex function on Riemannian manifold by use of Penot generalized directional derivative and the Clarke generalized gradient. This paper also presents a method for judging whether a point is the global minimum point in the inequality constraints. Our objective here is to extend the content and proof the necessary and sufficient condition of convex function to Riemannian manifolds.
Linear approximation of the first eigenvalue on compact manifolds
Institute of Scientific and Technical Information of China (English)
CHEN; Mufa(陈木法); E.; Scacciatelli; YAO; Liang(姚亮)
2002-01-01
For compact, connected Riemannian manifolds with Ricci curvature bounded below by a constant, what is the linear approximation of the first eigenvalue of Laplacian? The answer is presented with computer assisted proof and the result is optimal in certain sense.
Dynamical systems on a Riemannian manifold that admit normal shift
Energy Technology Data Exchange (ETDEWEB)
Boldin, A.Yu.; Dmitrieva, V.V.; Safin, S.S.; Sharipov, R.A. [Bashkir State Univ. (Russian Federation)
1995-11-01
Newtonian dynamical systems that admit normal shift on an arbitrary Riemannian manifold are considered. The determining equations for these systems, which constitute the condition of weak normality, are derived. The extension of the algebra of tensor fields to manifolds is considered.
4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings
Davis, M; Lafont, J -F
2010-01-01
We construct examples of smooth 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at infinity of F defines a nontrivial knot in the boundary at infinity of X. As a consequence, we obtain that the fundamental group of M cannot be isomorphic to the fundamental group of any Riemannian manifold of nonpositive sectional curvature. In particular, M is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.
Dirac structures on generalized Riemannian manifolds
Vaisman, Izu
2011-01-01
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, have a symplectic structure. Particularly, if compatibility with a generalized Riemannian metric is required, the symplectic structure is of the Kaehler type.
Riemannian manifolds as Lie-Rinehart algebras
Pessers, Victor; van der Veken, Joeri
2016-07-01
In this paper, we show how Lie-Rinehart algebras can be applied to unify and generalize the elementary theory of Riemannian geometry. We will first review some necessary theory on a.o. modules, bilinear forms and derivations. We will then translate some classical theory on Riemannian geometry to the setting of Rinehart spaces, a special kind of Lie-Rinehart algebras. Some generalized versions of classical results will be obtained, such as the existence of a unique Levi-Civita connection, inducing a Levi-Civita connection on a submanifold, and the construction of spaces with constant sectional curvature.
Almost conformal transformation in a class of Riemannian manifolds
Dzhelepov, Georgi; Dokuzova, Iva
2010-01-01
We consider a 3-dimensional Riemannian manifold V with a metric g and an affinor structure q. The local coordinates of these tensors are circulant matrices. In V we define an almost conformal transformation. Using that definition we construct an infinite series of circulant metrics which are successively almost conformaly related. In this case we get some properties.
On the de Rham-Wu decomposition for Riemannian and Lorentzian manifolds
Galaev, Anton S
2016-01-01
It is explained how to find the de~Rham decomposition of a Riemannian manifold and the Wu decomposition of a Lorentzian manifold. For that it is enough to find parallel symmetric bilinear forms on the manifold, and do some linear algebra. This result will allow to compute the connected holonomy group of an arbitrary Riemannian or Lorentzian manifold.
Wind Turbine Gearbox Fault Diagnosis Method Based on Riemannian Manifold
Shoubin Wang; Xiaogang Sun; Chengwei Li
2014-01-01
As multivariate time series problems widely exist in social production and life, fault diagnosis method has provided people with a lot of valuable information in the finance, hydrology, meteorology, earthquake, video surveillance, medical science, and other fields. In order to find faults in time sequence quickly and efficiently, this paper presents a multivariate time series processing method based on Riemannian manifold. This method is based on the sliding window and uses the covariance mat...
Invariant tensors related with natural connections for a class Riemannian product manifolds
Gribacheva, Dobrinka
2012-01-01
Some invariant tensors in two Naveira classes of Riemannian product manifolds are considered. These tensors are related with natural connections, i.e. linear connections preserving the Riemannian metric and the product structure.
Elliptic Equations in Weighted Besov Spaces on Asymptotically Flat Riemannian Manifolds
Brauer, Uwe
2012-01-01
This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptotically flat Riemannian manifolds, and in particular to the solutions of Einstein's constraints equations. We establish existence theorems for the Hamiltonian an momentum constraints with constant mean curvature and with a background metric which satisfies very low regularity assumptions. These results extend the regularity results of Holst, Nagy and Tsogtgerel about the constraint equations on compact manifolds in the Besov space $B_{p,p}^s$, to asymptotically flat manifolds. We also consider the Brill--Cantor criterion in the weighted Besov spaces. Our results improve the regularity assumptions on asymptotically flat manifolds Choquet--Bruhat, Isenberg and Pollack, and Maxwell, as well as they enable us to construct the initial data for the Einstein--Euler system.
Nonlinear damped oscillators on Riemannian manifolds: Numerical simulation
Fiori, Simone
2017-06-01
Nonlinear oscillators are ubiquitous in sciences, being able to model the behavior of complex nonlinear phenomena, as well as in engineering, being able to generate repeating (i.e., periodic) or non-repeating (i.e., chaotic) reference signals. The state of the classical oscillators known from the literature evolves in the space Rn , typically with n = 1 (e.g., the famous van der Pol vacuum-tube model), n = 2 (e.g., the FitzHugh-Nagumo model of spiking neurons) or n = 3 (e.g., the Lorenz simplified model of turbulence). The aim of the current paper is to present a general scheme for the numerical differential-geometry-based integration of a general second-order, nonlinear oscillator model on Riemannian manifolds and to present several instances of such model on manifolds of interest in sciences and engineering, such as the Stiefel manifold and the space of symmetric, positive-definite matrices.
Wind Turbine Gearbox Fault Diagnosis Method Based on Riemannian Manifold
Directory of Open Access Journals (Sweden)
Shoubin Wang
2014-01-01
Full Text Available As multivariate time series problems widely exist in social production and life, fault diagnosis method has provided people with a lot of valuable information in the finance, hydrology, meteorology, earthquake, video surveillance, medical science, and other fields. In order to find faults in time sequence quickly and efficiently, this paper presents a multivariate time series processing method based on Riemannian manifold. This method is based on the sliding window and uses the covariance matrix as a descriptor of the time sequence. Riemannian distance is used as the similarity measure and the statistical process control diagram is applied to detect the abnormity of multivariate time series. And the visualization of the covariance matrix distribution is used to detect the abnormity of mechanical equipment, leading to realize the fault diagnosis. With wind turbine gearbox faults as the experiment object, the fault diagnosis method is verified and the results show that the method is reasonable and effective.
A New Family of Curvature Homogeneous Pseudo-Riemannian Manifolds
Dunn, Corey
2009-01-01
We construct a new family of curvature homogeneous pseudo-Riemannian manifolds modeled on $\\mathbb{R}^{3k+2}$ for integers $k \\geq 1$. In contrast to previously known examples, the signature may be chosen to be $(k+1+a, k+1+b)$ where $a,b \\in \\mathbb{N} \\bigcup \\{0\\}$ and $a+b = k$. The structure group of the 0-model of this family is studied, and is shown to be indecomposable. Several invariants that are not of Weyl type are found which will show that, in general, the members of this family ...
Eigenvalues of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds
Directory of Open Access Journals (Sweden)
Giovanni Rastelli
2007-02-01
Full Text Available Given a $n$-dimensional Riemannian manifold of arbitrary signature, we illustrate an algebraic method for constructing the coordinate webs separating the geodesic Hamilton-Jacobi equation by means of the eigenvalues of $m leq n$ Killing two-tensors. Moreover, from the analysis of the eigenvalues, information about the possible symmetries of the web foliations arises. Three cases are examined: the orthogonal separation, the general separation, including non-orthogonal and isotropic coordinates, and the conformal separation, where Killing tensors are replaced by conformal Killing tensors. The method is illustrated by several examples and an application to the L-systems is provided.
Stretch fast dynamo mechanism via conformal mapping in Riemannian manifolds
Garcia de Andrade, L. C.
2007-10-01
Two new analytical solutions of the self-induction equation in Riemannian manifolds are presented. The first represents a twisted magnetic flux tube or flux rope in plasma astrophysics, where the rotation of the flow implies that the poloidal field is amplified from toroidal field, in the spirit of dynamo theory. The value of the amplification depends on the Frenet torsion of the magnetic axis of the tube. Actually this result illustrates the Zeldovich stretch, twist, and fold method to generate dynamos from straight and untwisted ropes. Based on the fact that this problem was previously handled, using a Riemannian geometry of twisted magnetic flux ropes [Phys Plasmas 13, 022309 (2006)], investigation of a second dynamo solution, conformally related to the Arnold kinematic fast dynamo, is obtained. In this solution, it is shown that the conformal effect on the fast dynamo metric enhances the Zeldovich stretch, and therefore a new dynamo solution is obtained. When a conformal mapping is performed in an Arnold fast dynamo line element, a uniform stretch is obtained in the original line element.
Near-equality of the Penrose Inequality for rotationally symmetric Riemannian manifolds
Lee, Dan A
2011-01-01
This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries are outermost minimal hypersurfaces. Specifically, we prove that if the Penrose Inequality is sufficiently close to being an equality on one of these manifolds, then it must be close to a Schwarzschild space with an appended cylinder, in the sense of Lipschitz Distance. Since the Lipschitz Distance bounds the Intrinsic Flat Distance on compact sets, we also obtain a result for Intrinsic Flat Distance, which is a more appropriate distance for more general near-equality results, as discussed in [LS
Rigid supersymmetry on 5-dimensional Riemannian manifolds and contact geometry
Energy Technology Data Exchange (ETDEWEB)
Pan, Yiwen [C.N. Yang Institute for Theoretical Physics,Stony Brook, NY, 11790 (United States)
2014-05-12
In this note we generalize the methods of http://dx.doi.org/10.1007/JHEP08(2012)141, http://dx.doi.org/10.1007/JHEP01(2013)072 and http://dx.doi.org/10.1007/JHEP05(2013)017 to 5-dimensional Riemannian manifolds M. We study the relations between the geometry of M and the number of solutions to a generalized Killing spinor equation obtained from a 5-dimensional supergravity. The existence of 1 pair of solutions is related to almost contact metric structures. We also discuss special cases related to M=S{sup 1}×M{sub 4}, which leads to M being foliated by submanifolds with special properties, such as Quaternion-Kähler. When there are 2 pairs of solutions, the closure of the isometry sub-algebra generated by the solutions requires M to be S{sup 3} or T{sup 3}-fibration over a Riemann surface. 4 pairs of solutions pin down the geometry of M to very few possibilities. Finally, we propose a new supersymmetric theory for N=1 vector multiplet on K-contact manifold admitting solutions to the Killing spinor equation.
Lattice Dirac Fermions on a Simplicial Riemannian Manifold
Brower, Richard C; Gasbarro, Andrew D; Raben, Timothy G; Tan, Chung-I; Weinberg, Evan S
2016-01-01
The lattice Dirac equation is formulated on a simplicial complex which approximates a smooth Riemann manifold by introducing a lattice vierbein on each site and a lattice spin connection on each link. Care is taken so the construction applies to any smooth D-dimensional Riemannian manifold that permits a spin connection. It is tested numerically in 2D for the projective sphere ${\\mathbb S}^2$ in the limit of an increasingly refined sequence of triangles. The eigenspectrum and eigenvectors are shown to converge rapidly to the exact result in the continuum limit. In addition comparison is made with the continuum Ising conformal field theory on ${\\mathbb S}^2$. Convergence is tested for the two point, $\\langle \\epsilon(x_1) \\epsilon(x_2) \\rangle$, and the four point, $\\langle \\sigma(x_1) \\epsilon(x_2) \\epsilon(x_3 )\\sigma(x_4) \\rangle $, correlators for the energy, $\\epsilon(x) = i \\bar \\psi(x)\\psi(x)$, and twist operators, $\\sigma(x)$, respectively.
Directory of Open Access Journals (Sweden)
Feng Qi
2014-10-01
Full Text Available The authors find the absolute monotonicity and complete monotonicity of some functions involving trigonometric functions and related to estimates the lower bounds of the first eigenvalue of Laplace operator on Riemannian manifolds.
Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds
Rosales, César
2010-01-01
For constant mean curvature surfaces of class $C^2$ immersed inside Sasakian sub-Riemannian $3$-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classification results for complete surfaces with empty singular set which are stable, i.e., second order minima of the area under a volume constraint, inside the $3$-dimensional sub-Riemannian space forms. In the first Heisenberg group we show that such a surface is a vertical plane. In the sub-Riemannian hyperbolic $3$-space we give an upper bound for the mean curvature of such surfaces, and we characterize the horocylinders as the only ones with squared mean curvature $1$. Finally we deduce that any complete surface with empty singular set in the sub-Riemannian $3$-sphere is unstable.
Directory of Open Access Journals (Sweden)
Erol Kılıç
2012-01-01
Full Text Available We study lightlike hypersurfaces of a semi-Riemannian product manifold. We introduce a class of lightlike hypersurfaces called screen semi-invariant lightlike hypersurfaces and radical anti-invariant lightlike hypersurfaces. We consider lightlike hypersurfaces with respect to a quarter-symmetric nonmetric connection which is determined by the product structure. We give some equivalent conditions for integrability of distributions with respect to the Levi-Civita connection of semi-Riemannian manifolds and the quarter-symmetric nonmetric connection, and we obtain some results.
On f-Eikonal Helices And f-Eikonal Slant Helices In Riemannian Manifolds
Şenol, Ali; Ziplar, Evren; Yayli, Yusuf
2012-01-01
In this paper, we define f-eikonal helix curves and f-eikonal V_{n}-slant helix curves in a n-dimensional Riemannian manifold. Also, we give the definition of harmonic curvature functions related to f-eikonal helix curves and f-eikonal V_{n}-slant helix curves in a n-dimensional Riemannian manifold. Moreover, we give characterizations for f-eikonal helix curves and f-eikonal V_{n}-slant helix curves by making use of the harmonic curvature functions.
A natural connection on a basic class of Riemannian product manifolds
Gribacheva, Dobrinka
2011-01-01
A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M; P; g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kahler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M; P; g) (i.e. DP = Dg = 0). We find necessary and suffcient conditions the curvature tensor of D to have properties similar to the Kahler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion.We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a at connection. We construct an example of the considered manifold by a Lie group where D is a...
Complete Formulas for the Volumes of Tubes about Curves in a Riemannian Manifold
Institute of Scientific and Technical Information of China (English)
Fa En WU
2001-01-01
By using the Taylor expansions of the solutions of Jacobi equations, we obtain the completeformulas for the volumes of tubes about curves in a Riemannian manifold. This unifies the known resultsand simplifies the computations involved in this direction. In the special case of surfaces, we also obtainthe corresponding complete formulas which generalize the known results.
The Degree of Symmetry of Certain Compact Smooth Manifolds Ⅱ
Institute of Scientific and Technical Information of China (English)
Bin XU
2007-01-01
We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2 × V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.
Flows Associated to Cameron-martin Type Vector Fields on Path Spaces Over a Riemannian Manifold
Institute of Scientific and Technical Information of China (English)
Jing-xiao ZHANG
2013-01-01
The flow on the Wiener space associated to a tangent process constructed by Cipriano and Cruzeiro,as well as by Gong and Zhang does not allow to recover Driver's Cameron-Martin theorem on Riemannian path space.The purpose of this work is to refine the method of the modified Picard iteration used in the previous work by Gong and Zhang and to try to recapture and extend the result of Driver.In this paper,we establish the existence and uniqueness of a flow associated to a Cameron-Martin type vector held on the path space over a Riemannian manifold.
Connection with Totally Skew-Symmetric Torsion on Riemannian Almost Product Manifolds
Mekerov, Dimitar
2010-01-01
On Riemannian almost product manifolds (M,P,g) with trP=0 we consider a linear connection nabla' preserving the almost product structure P and the metric g and having a totally skew-symmetric torsion tensor. We prove that if (M,P,g) admits such a connection then (M,P,g) belongs to the basic class W3 from the classification in [M.Staikova, K.Gribachev. Canonical connections and their conformal invariants on Riemannian P-manifolds, Serdica Math. J. 18 (1992), 150-161]. We consider the case when the curvature tensor of nabla' on (M,P,g) in W3 has the same properties like the ones of the curvature tensor of the Levi-Civita connection nabla on (M,P,g) with nabla(P)=0. We construct a 4-parametric family of 4-dimensional Riemannian manifolds of the class W3 by a Lie group and consider the connection nabla' on an arbitrary manifold of this family.
Directory of Open Access Journals (Sweden)
Fan Yang
2015-07-01
Full Text Available Normally, polarimetric SAR classification is a high-dimensional nonlinear mapping problem. In the realm of pattern recognition, sparse representation is a very efficacious and powerful approach. As classical descriptors of polarimetric SAR, covariance and coherency matrices are Hermitian semidefinite and form a Riemannian manifold. Conventional Euclidean metrics are not suitable for a Riemannian manifold, and hence, normal sparse representation classification cannot be applied to polarimetric SAR directly. This paper proposes a new land cover classification approach for polarimetric SAR. There are two principal novelties in this paper. First, a Stein kernel on a Riemannian manifold instead of Euclidean metrics, combined with sparse representation, is employed for polarimetric SAR land cover classification. This approach is named Stein-sparse representation-based classification (SRC. Second, using simultaneous sparse representation and reasonable assumptions of the correlation of representation among different frequency bands, Stein-SRC is generalized to simultaneous Stein-SRC for multi-frequency polarimetric SAR classification. These classifiers are assessed using polarimetric SAR images from the Airborne Synthetic Aperture Radar (AIRSAR sensor of the Jet Propulsion Laboratory (JPL and the Electromagnetics Institute Synthetic Aperture Radar (EMISAR sensor of the Technical University of Denmark (DTU. Experiments on single-band and multi-band data both show that these approaches acquire more accurate classification results in comparison to many conventional and advanced classifiers.
The geometric median on Riemannian manifolds with application to robust atlas estimation.
Fletcher, P Thomas; Venkatasubramanian, Suresh; Joshi, Sarang
2009-03-01
One of the primary goals of computational anatomy is the statistical analysis of anatomical variability in large populations of images. The study of anatomical shape is inherently related to the construction of transformations of the underlying coordinate space, which map one anatomy to another. It is now well established that representing the geometry of shapes or images in Euclidian spaces undermines our ability to represent natural variability in populations. In our previous work we have extended classical statistical analysis techniques, such as averaging, principal components analysis, and regression, to Riemannian manifolds, which are more appropriate representations for describing anatomical variability. In this paper we extend the notion of robust estimation, a well established and powerful tool in traditional statistical analysis of Euclidian data, to manifold-valued representations of anatomical variability. In particular, we extend the geometric median, a classic robust estimator of centrality for data in Euclidean spaces. We formulate the geometric median of data on a Riemannian manifold as the minimizer of the sum of geodesic distances to the data points. We prove existence and uniqueness of the geometric median on manifolds with non-positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds. Generalizing the Weiszfeld procedure for finding the geometric median of Euclidean data, we present an algorithm for computing the geometric median on an arbitrary manifold. We show that this algorithm converges to the unique solution when it exists. In this paper we exemplify the robustness of the estimation technique by applying the procedure to various manifolds commonly used in the analysis of medical images. Using this approach, we also present a robust brain atlas estimation technique based on the geometric median in the space of deformable images.
Convergence of inexact descent methods for nonconvex optimization on Riemannian manifolds
Bento, G C; Oliveira, P R
2011-01-01
In this paper we present an abstract convergence analysis of inexact descent methods in Riemannian context for functions satisfying Kurdyka-Lojasiewicz inequality. In particular, without any restrictive assumption about the sign of the sectional curvature of the manifold, we obtain full convergence of a bounded sequence generated by the proximal point method, in the case that the objective function is nonsmooth and nonconvex, and the subproblems are determined by a quasi distance which does not necessarily coincide with the Riemannian distance. Moreover, if the objective function is $C^1$ with $L$-Lipschitz gradient, not necessarily convex, but satisfying Kurdyka-Lojasiewicz inequality, full convergence of a bounded sequence generated by the steepest descent method is obtained.
Self-adjoint Extensions of Schrödinger Operators with ?-magnetic Fields on Riemannian Manifolds
Directory of Open Access Journals (Sweden)
T. Mine
2010-01-01
Full Text Available We consider the magnetic Schr¨odinger operator on a Riemannian manifold M. We assume the magnetic field is given by the sum of a regular field and the Dirac δ measures supported on a discrete set Γ in M. We give a complete characterization of the self-adjoint extensions of the minimal operator, in terms of the boundary conditions. The result is an extension of the former results by Dabrowski-Šťoviček and Exner-Šťoviček-Vytřas.
Dokuzova, Iva
2010-01-01
In the present paper it is considered a class V of 3-dimensional Riemannian manifolds M with a metric g and two affinor tensors q and S. It is defined another metric \\bar{g} in M. The local coordinates of all these tensors are circulant matrices. It is found: 1)\\ a relation between curvature tensors R and \\bar{R} of g and \\bar{g}, respectively; 2)\\ an identity of the curvature tensor R of g in the case when the curvature tensor \\bar{R} vanishes; 3)\\ a relation between the sectional curvature of a 2-section of the type \\{x, qx\\} and the scalar curvature of M.
Eikonal slant helices and eikonal Darboux helices in 3-dimensional pseudo-Riemannian manifolds
Önder, Mehmet; Ziplar, Evren
2013-01-01
In this study, we give definitions and characterizations of eikonal slant helices, eikonal Darboux helices and non-normed eikonal Darboux helices in 3-dimensional pseudo- Riemannian manifold M . We show that every eikonal slant helix is also an eikonal Darboux helix for timelike and spacelike curves. Furthermore, we obtain that if the non-null curve a is a non-normed eikonal Darboux helix, then a is an eikonal slant helix if and only if 2 2 e 3k +e1t = constant, where k and t are curvature an...
Eikonal Slant Helices and Eikonal Darboux Helices In 3-Dimensional Riemannian Manifolds
Önder, Mehmet; Ziplar, Evren; Kaya, Onur
2013-01-01
In this study, we give definitions and characterizations of eikonal slant helix curves, eikonal Darboux helices and non-normed eikonal Darboux helices in three dimensional Riemannian manifold 3 M . We show that every eikonal slant helix is also an eikonal Darboux helix. Furthermore, we obtain that if the curve a is a non-normed eikonal Darboux helix, then a is an eikonal slant helix if and only if k 2 +t 2 = constant, where k and t are curvature and torsion of a, respectively.
Renormalization Proof for Massive $\\vp_4^4$ Theory on Riemannian Manifolds
Kopper, C
2006-01-01
In this paper we present an inductive renormalizability proof for massive $\\vp_4^4$ theory on Riemannian manifolds, based on the Wegner-Wilson flow equations of the Wilson renormalization group, adapted to perturbation theory. The proof goes in hand with bounds on the perturbative Schwinger functions which imply tree decay between their position arguments. An essential prerequisite are precise bounds on the short and long distance behaviour of the heat kernel on the manifold. With the aid of a regularity assumption (often taken for granted) we also show, that for suitable renormalization conditions the bare action takes the minimal form, that is to say, there appear the same counter terms as in flat space, apart from a logarithmically divergent one which is proportional to the scalar curvature.
Einstein constraints on n dimensional compact manifolds
Choquet-Bruhat, Y
2004-01-01
We give a general survey of the solution of the Einstein constraints by the conformal method on n dimensional compact manifolds. We prove some new results about solutions with low regularity (solutions in $H_{2}$ when n=3), and solutions with unscaled sources.
Localization of supersymmetric field theories on non-compact hyperbolic three-manifolds
Assel, Benjamin; Murthy, Sameer; Yokoyama, Daisuke
2016-01-01
We study supersymmetric gauge theories with an R-symmetry, defined on non-compact, hyperbolic, Riemannian three-manifolds, focusing on the case of a supersymmetry-preserving quotient of Euclidean AdS$_3$. We compute the exact partition function in these theories, using the method of localization, thus reducing the problem to the computation of one-loop determinants around a supersymmetric locus. We evaluate the one-loop determinants employing three different techniques: an index theorem, the method of pairing of eigenvalues, and the heat kernel method. Along the way, we discuss aspects of supersymmetry in manifolds with a conformal boundary, including supersymmetric actions and boundary conditions.
DEFF Research Database (Denmark)
Zimmermann, Ralf
2017-01-01
We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the optimization-based approach known from the literature, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm...... converges locally and exhibits a linear rate of convergence....
Giambo', R; Piccione, P
2010-01-01
In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for a class of Hamiltonian systems at a fixed energy level.
DEFF Research Database (Denmark)
Zimmermann, Ralf
2017-01-01
We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the optimization-based approach known from the literature, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm...... converges locally and exhibits a linear rate of convergence....
Spacelike Graphs with Parallel Mean Curvature in Pseudo-Riemannian Product Manifolds
Institute of Scientific and Technical Information of China (English)
Zicheng ZHAO
2012-01-01
The author introduces the w-function defined on the considered spacelike graph M.Under the growth conditions w =o(log z) and w =o(r),two Bernstein type theorems for M in Rmn+m are got,where z and r are the pseudo-Euclidean distance and the distance function on M to some fixed point respectively.As the ambient space is a curved pseudoRiemannian product of two Riemannian manifolds (∑1,g1) and (∑2,g2) of dimensions n and m,a Bernstein type result for n =2 under some curvature conditions on ∑1 and ∑2 and the growth condition w =o(r) is also got.As more general cases,under some curvature conditions on the ambient space and the growth condition w =o(Υ) or w =o(√Υ),the author concludes that if M has parallel mean curvature,then M is maximal.
Dokuzova, Iva
2011-01-01
We consider a four dimensional Riemannian manifold M with a metric g and affinor structure q. The local coordinates of these tensors are circulant matrices. Their first orders are (A, B, C, B), A, B, C\\in FM and (0, 1, 0, 0), respectively. We construct another metric \\tilde{g} on M. We find the conditions for \\tilde{g} to be a positively defined metric, and for q to be a parallel structure with respect to the Riemannian connection of g. Further, let x be an arbitrary vector in T_{p}M, where p is a point on M. Let \\phi and \\phi be the angles between x and qx, x and q^{2}x with respect to g. We express the angles between x and qx, x and q^{2}x with respect to $\\tilde{g}$ with the help of the angles $\\phi$ and \\phi. Also,we construct two series {\\phi_{n}}and {\\phi_{n}}. We prove that every of it is an increasing one and it is converge.
Geodesic B-Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds
Directory of Open Access Journals (Sweden)
Sheng-lan Chen
2014-01-01
Full Text Available We introduce a class of functions called geodesic B-preinvex and geodesic B-invex functions on Riemannian manifolds and generalize the notions to the so-called geodesic quasi/pseudo B-preinvex and geodesic quasi/pseudo B-invex functions. We discuss the links among these functions under appropriate conditions and obtain results concerning extremum points of a nonsmooth geodesic B-preinvex function by using the proximal subdifferential. Moreover, we study a differentiable multiobjective optimization problem involving new classes of generalized geodesic B-invex functions and derive Kuhn-Tucker-type sufficient conditions for a feasible point to be an efficient or properly efficient solution. Finally, a Mond-Weir type duality is formulated and some duality results are given for the pair of primal and dual programming.
Riesz-Martin representation for positive super-polyharmonic functions in A Riemannian manifold
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V. Anandam
2006-07-01
Full Text Available Let u be a super-biharmonic function, that is, ÃŽÂ”2uÃ¢Â‰Â¥0, on the unit disc D in the complex plane, satisfying certain conditions. Then it has been shown that u has a representation analogous to the Poisson-Jensen representation for subharmonic functions on D. In the same vein, it is shown here that a function u on any Green domain ÃŽÂ© in a Riemannian manifold satisfying the conditions (Ã¢ÂˆÂ’ÃŽÂ”iuÃ¢Â‰Â¥0 for 0Ã¢Â‰Â¤iÃ¢Â‰Â¤m has a representation analogous to the Riesz-Martin representation for positive superharmonic functions on ÃŽÂ©.
Isometry group and geodesics of the Wagner lift of a riemannian metric on two-dimensional manifold
B., José Ricardo Arteaga
2010-01-01
In this paper we construct a functor from the category of two-dimensional Riemannian manifolds to the category of three-dimensional manifolds with generalized metric tensors. For each two-dimensional oriented Riemannian manifold $(M,g)$ we construct a metric tensor $\\hat g$ (in general, with singularities) on the total space $SO(M,g)$ of the principal bundle of the positively oriented orthonormal frames on $M$. We call the metric $\\hat g$ the Wagner lift of $g$. We study the relation between the isometry groups of $(M,g)$ and $(SO(M,g),\\hat g)$. We prove that the projections of the geodesics of $(SO(M,g),\\hat g)$ onto $M$ are the curves which satisfy the equation \\begin{equation*} \
Seiberg-Witten Like Equations on Pseudo-Riemannian Spinc Manifolds with G2(2∗ Structure
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Nülifer Özdemir
2016-01-01
Full Text Available We consider 7-dimensional pseudo-Riemannian spinc manifolds with structure group G2(2∗. On such manifolds, the space of 2-forms splits orthogonally into components Λ2M=Λ72⊕Λ142. We define self-duality of a 2-form by considering the part Λ72 as the bundle of self-dual 2-forms. We express the spinor bundle and the Dirac operator and write down Seiberg-Witten like equations on such manifolds. Finally we get explicit forms of these equations on R4,3 and give some solutions.
Gooya, Ali; Liao, Hongen; Sakuma, Ichiro
2012-09-01
Geometric flux maximizing flow (FLUX) is an active contour based method which evolves an initial surface to maximize the flux of a vector field on the surface. For blood vessel segmentation, the vector field is defined as the vectors specified by vascular edge strengths and orientations. Hence, the segmentation performance depends on the quality of the detected edge vector field. In this paper, we propose a new method for level set based segmentation of blood vessels by generalizing the FLUX on a Riemannian manifold (R-FLUX). We consider a 3D scalar image I(x) as a manifold embedded in the 4D space (x, I(x)) and compute the image metric by pullback from the 4D space, whose metric tensor depends on the vessel enhancing diffusion (VED) tensor. This allows us to devise a non-linear filter which both projects and normalizes the original image gradient vectors under the inverse of local VED tensors. The filtered gradient vectors pertaining to the vessels are less sensitive to the local image contrast and more coherent with the local vessel orientation. The method has been applied to both synthetic and real TOF MRA data sets. Comparisons are made with the FLUX and vesselsness response based segmentations, indicating that the R-FLUX outperforms both methods in terms of leakage minimization and thiner vessel delineation.
ASYMPTOTIC LINKING INVARIANTS FOR RKACTIONS IN COMPACT RIEMANNIAN MANIFOLDS
JOSE LUIS LIZARBE CHIRA
2005-01-01
Arnold no seu trabalho The asymptotic Hopf Invariant and its applications de 1986, considerou sobre um domínio (ômega maiúsculo) compacto de R3 com bordo suave e homología trivial campos X e Y de divergência nula e tangentes ao bordo de (ômega maiúsculo) e definiu o índice de enlaçamento assintótico lk(X; Y ) e o invariante de Hopf associados a X e Y pela integral I(X; Y ) = (integral em ômega maiúsculo de alfa produto d- beta), onde (d-alfa) = iX-vol e (d-b...
Directory of Open Access Journals (Sweden)
Andrei V. Obukhovskiĭ
2003-05-01
Full Text Available We consider second-order differential inclusions on a Riemannian manifold with lower semicontinuous right-hand sides. Several existence theorems for solutions of two-point boundary value problem are proved to be interpreted as controllability of special mechanical systems with control on nonlinear configuration spaces. As an application, a statement of controllability under extreme values of controlling force is obtained.
Lefschetz Fibrations on Compact Stein Manifolds
Akbulut, Selman
2010-01-01
Here we prove that a compact Stein manifold W of dimension 2n+2>4 admits a Lefschetz fibration over the 2-disk with Stein fibers, such that the monodromy of the fibration is a symplectomorphism induced by compositions of "generalized Dehn twists" along imbedded n-spheres on the generic fiber. Also, the open book on the boundary of W, which is determined by the fibration, is compatible with the contact structure induced by the Stein structure. This generalizes the Stein surface case of n=1, previously proven by Loi-Piergallini and Akbulut-Ozbagci.
Absence of embedded eigenvalues for Riemannian Laplacians
DEFF Research Database (Denmark)
Ito, Kenichi; Skibsted, Erik
Schrödinger operators on non-compact connected Riemannian manifolds. A principal example is given by a manifold with an end (possibly more than one) in which geodesic coordinates are naturally defined. In this case one of our geometric conditions is a positive lower bound of the second fundamenta...
Ibort, A
2012-01-01
In these three lectures we will discuss some fundamental aspects of the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on compact Riemannian manifolds with smooth boundary emphasizing the relation with the theory of global boundary conditions. Self-adjoint extensions of symmetric operators, specially of the Laplace-Beltrami and Dirac operators, are fundamental in Quantum Physics as they determine either the energy of quantum systems and/or their unitary evolution. The well-known von Neumann's theory of self-adjoint extensions of symmetric operators is not always easily applicable to differential operators, while the description of extensions in terms of boundary conditions constitutes a more natural approach. Thus an effort is done in offering a description of self-adjoint extensions in terms of global boundary conditions showing how an important family of self-adjoint extensions for the Laplace-Beltrami and Dirac operators are easily describable in this way. Moreover ...
Local Schrodinger flow into Kahler manifolds
Institute of Scientific and Technical Information of China (English)
丁伟岳; 王友德
2001-01-01
In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schrodinger flow for maps from a compact Riemannian manifold into a complete Kahler manifold, or from a Euclidean space Rm into a compact Kahler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.
Finite Time and Exact Time Controllability on Compact Manifolds
Jouan, Philippe
2010-01-01
It is first shown that a smooth controllable system on a compact manifold is finite time controllable. The technique of proof is close to the one of Sussmann's orbit theorem, and no rank condition is required. This technique is also used to give a new and elementary proof of the equivalence between controllability for essentially bounded inputs and for piecewise constant ones. Two sufficient conditions for controllability at exact time on a compact manifold are then stated. Some applications,...
Automorphisms and examples of compact non K\\"ahler manifolds
Magnússon, Gunnar Þór
2012-01-01
Let $X$ be a compact K\\"ahler manifold with zero first Chern class and finite fundamental group. Folklore says that if an automorphism $f$ of $X$ fixes a K\\"ahler class, then its order is finite. We apply this result to construct a compact non K\\"ahler manifold $F$ as a fibration $X \\to F \\to B$ over a complex torus $B$.
Institute of Scientific and Technical Information of China (English)
刘建成; 柴瑞娟
2013-01-01
设(M,g)是一个具有Killing向量场ε的伪黎曼流形.本文讨论M的紧致类空超曲面M上Killing向量场的存在性问题,给出了M上存在Killing向量场的一个充分条件.%Let(（M）,g)be a semi-Riemannian manifold with a Killing vector field ε.In this paper,the existence problems of Killing vector fields on compact spacelike hypersurface M of M are studied,and a sufficient condition for the existence of Killing vector fields is obtained.
Dynamics and zeta functions on conformally compact manifolds
Rowlett, Julie; Tapie, Samuel
2011-01-01
In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces, we obtain optimal meromorphic extensions of weighted dynamical zeta functions and asymptotic counting estimates for the number of weighted closed geodesics. A meromorphic extension of the standard dynamical zeta function and the prime orbit theorem follow as corollaries. Finally, we investigate interactions between the dynamics and spectral theory of these spaces.
On the geometry of Riemannian manifolds with a Lie structure at infinity
Directory of Open Access Journals (Sweden)
Bernd Ammann
2004-01-01
Full Text Available We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.
Segmentation of high angular resolution diffusion MRI using sparse riemannian manifold clustering.
Çetingül, H Ertan; Wright, Margaret J; Thompson, Paul M; Vidal, René
2014-02-01
We address the problem of segmenting high angular resolution diffusion imaging (HARDI) data into multiple regions (or fiber tracts) with distinct diffusion properties. We use the orientation distribution function (ODF) to model diffusion and cast the ODF segmentation problem as a clustering problem in the space of ODFs. Our approach integrates tools from sparse representation theory and Riemannian geometry into a graph theoretic segmentation framework. By exploiting the Riemannian properties of the space of ODFs, we learn a sparse representation for each ODF and infer the segmentation by applying spectral clustering to a similarity matrix built from these representations. In cases where regions with similar (resp. distinct) diffusion properties belong to different (resp. same) fiber tracts, we obtain the segmentation by incorporating spatial and user-specified pairwise relationships into the formulation. Experiments on synthetic data evaluate the sensitivity of our method to image noise and to the concentration parameters, and show its superior performance compared to alternative methods when analyzing complex fiber configurations. Experiments on phantom and real data demonstrate the accuracy of the proposed method in segmenting simulated fibers and white matter fiber tracts of clinical importance.
Invariant measures and controllability of finite systems on compact manifolds
Jouan, Philippe
2010-01-01
A control system is said to be finite if the Lie algebra generated by its vector fields is finite dimensional. Sufficient conditions for such a system on a compact manifold to be controllable are stated in terms of its Lie algebra. The proofs make use of the Equivalence Theorem of \\cite{Jouan09} and of the existence of an invariant measure on certain compact homogeneous spaces.
Besov continuity for pseudo-differential operators on compact homogeneous manifolds
Cardona, Duván
2016-01-01
In this paper we study the Besov continuity of pseudo-differential operators on compact homogeneous manifolds $M=G/K.$ We use the global quantization of these operators in terms of the representation theory of compact homogeneous manifolds.
Petersen, Peter
2016-01-01
Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups. Important revisions to the third edition include: a substantial addition of unique and enriching exercises scattered throughout the text; inclusion of an increased number of coordinate calculations of connection and curvature; addition of general formulas for curvature on Lie Groups and submersions; integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger; incorporation of several recent results about manifolds with posit...
Einstein Metrics, Four-Manifolds, and Differential Topology
2004-01-01
This article presents a new and more elementary proof of the main Seiberg-Witten-based obstruction to the existence of Einstein metrics on smooth compact 4-manifolds. It also introduces a new smooth manifold invariant which conveniently encapsulates those aspects of Seiberg-Witten theory most relevant to the study of Riemannian variational problems on 4-manifolds.
Moduli spaces of riemannian metrics
Tuschmann, Wilderich
2015-01-01
This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.
Compactness of $\\Box_b$ in a CR manifold
Khanh, Tran Vu; Zampieri, Giuseppe
2011-01-01
This note is aimed at simplifying current literature about compactness estimates for the Kohn-Laplacian on CR manifolds. The approach consists in a tangential basic estimate in the formulation given by the first author in \\cite{Kh10} which refines former work by Nicoara \\cite{N06}. It has been proved by Raich \\cite{R10} that on a CR manifold of dimension $2n-1$ which is compact pseudoconvex of hypersurface type embedded in $\\C^n$ and orientable, the property named "$(CR-P_q)$" for $1\\leq q\\leq \\frac{n-1}2$, a generalization of the one introduced by Catlin in \\cite{C84}, implies compactness estimates for the Kohn-Laplacian $\\Box_b$ in degree $k$ for any $k$ satisfying $q\\leq k\\leq n-1-q$. The same result is stated by Straube in \\cite{S10} without the assumption of orientability. We regain these results by a simplified method and extend the conclusions in two directions. First, the CR manifold is no longer required to be embedded. Second, when $(CR-P_q)$ holds for $q=1$ (and, in case $n=1$, under the additional...
Medians and means in Riemannian geometry: existence, uniqueness and computation
Arnaudon, Marc; Yang, Le
2011-01-01
This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, Fr\\'echet medians are considered. We prove their statistical consistency and give some quantitative estimations of their robustness with the aid of upper curvature bounds. We also show that, in compact Riemannian manifolds, the Fr\\'echet medians of generic data points are always unique. Stochastic and deterministic algorithms are proposed for computing Riemannian p-means. The rate of convergence and error estimates of these algorithms are also obtained. Finally, we apply the medians and the Riemannian geometry of Toeplitz covariance matrices to radar target detection.
Random Lie group actions on compact manifolds: a perturbative analysis
Sadel, Christian
2008-01-01
A random Lie group action on a compact manifold generates a discrete time Markov process. The main object of this paper is the evaluation of associated Birkhoff sums in a regime of weak, but sufficiently effective coupling of the randomness. This effectiveness is expressed in terms of random Lie algebra elements and replaces the transience or Furstenberg's irreducibility hypothesis in related problems. The Birkhoff sum of any given smooth function then turns out to be equal to its integral w.r.t. a unique smooth measure on the manifold up to errors of the order of the coupling constant. Applications to the theory of products of random matrices and a model of a disordered quantum wire are presented.
Random Riesz energies on compact K\\"{a}hler manifolds
Feng, Renjie
2011-01-01
This article determines the asymptotics of the expected Riesz s-energy of the zero set of a Gaussian random systems of polynomials of degree N as the degree N tends to infinity in all dimensions and codimensions. The asymptotics are proved more generally for sections of any positive line bundle over any compact Kaehler manifold. In comparison with the results on energies of zero sets in one complex dimension due to Qi Zhong (arXiv:0705.2000) (see also [arXiv:0705.2000]), the zero sets have higher energies than randomly chosen points in dimensions > 2 due to clumping of zeros.
The $\\alpha'$ Expansion On A Compact Manifold Of Exceptional Holonomy
Becker, Katrin; Witten, Edward
2014-01-01
In the approximation corresponding to the classical Einstein equations, which is valid at large radius, string theory compactification on a compact manifold $M$ of $G_2$ or $\\mathrm{Spin}(7)$ holonomy gives a supersymmetric vacuum in three or two dimensions. Do $\\alpha'$ corrections to the Einstein equations disturb this statement? Explicitly analyzing the leading correction, we show that the metric of $M$ can be adjusted to maintain supersymmetry. Beyond leading order, a general argument based on low energy effective field theory in spacetime implies that this is true exactly (not just to all finite orders in $\\alpha'$). A more elaborate field theory argument that includes the massive Kaluza-Klein modes matches the structure found in explicit calculations. In M-theory compactification on a manifold $M$ of $G_2$ or Spin(7) holonomy, similar results hold to all orders in the inverse radius of $M$ -- but not exactly. The classical moduli space of $G_2$ metrics on a manifold $M$ is known to be locally a Lagrangi...
Fibrations and globalizations of compact homogeneous CR-manifolds
Gilligan, B.; Huckleberry, Alan T.
2009-06-01
Fibration methods which were previously used for complex homogeneous spaces and CR-homogeneous spaces of special types [1]-[4] are developed in a general framework. These include the \\mathfrak{g}-anticanonical fibration in the CR-setting, which reduces certain considerations to the compact projective algebraic case, where a Borel-Remmert type splitting theorem is proved. This leads to a reduction to spaces homogeneous under actions of compact Lie groups. General globalization theorems are proved which enable one to regard a homogeneous CR-manifold as an orbit of a real Lie group in a complex homogeneous space of a complex Lie group. In the special case of CR-codimension at most two, precise classification results are proved and are applied to show that in most cases there exists such a globalization.
Fibrations and globalizations of compact homogeneous CR-manifolds
Energy Technology Data Exchange (ETDEWEB)
Gilligan, B [University of Regina, Regina (Canada); Huckleberry, Alan T [Ruhr-Universitaet Bochum, Mathematischer Institut, Bochum (Germany)
2009-06-30
Fibration methods which were previously used for complex homogeneous spaces and CR-homogeneous spaces of special types [1]-[4] are developed in a general framework. These include the g-anticanonical fibration in the CR-setting, which reduces certain considerations to the compact projective algebraic case, where a Borel-Remmert type splitting theorem is proved. This leads to a reduction to spaces homogeneous under actions of compact Lie groups. General globalization theorems are proved which enable one to regard a homogeneous CR-manifold as an orbit of a real Lie group in a complex homogeneous space of a complex Lie group. In the special case of CR-codimension at most two, precise classification results are proved and are applied to show that in most cases there exists such a globalization.
Entropy production of stationary diffusions on non-compact Riemannian manifolds
Institute of Scientific and Technical Information of China (English)
龚光鲁; 钱敏平
1997-01-01
The closed form of the entropy production of stationary diffusion processes with bounded Nelson’s current velocity is given.The limit of the entropy productions of a sequence of reflecting diffusions is also discussed.
The construction of periodic unfolding operators on some compact Riemannian manifolds
DEFF Research Database (Denmark)
Dobberschütz, Sören; Böhm, Michael
2014-01-01
The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the "flat" Euclidean space R n. In this paper, we present a generalization of the method of periodic unfolding applicable to struct...
Entropy-expansiveness of Geodesic Flows on Closed Manifolds without Conjugate Points
Institute of Scientific and Technical Information of China (English)
Fei LIU; Fang WANG
2016-01-01
In this article, we consider the entropy-expansiveness of geodesic flows on closed Rieman-nian manifolds without conjugate points. We prove that, if the manifold has no focal points, or if the manifold is bounded asymptote, then the geodesic flow is entropy-expansive. Moreover, for the compact oriented surfaces without conjugate points, we prove that the geodesic flows are entropy-expansive. We also give an estimation of distance between two positively asymptotic geodesics of an uniform visibility manifold.
Learning in Riemannian Orbifolds
Jain, Brijnesh J
2012-01-01
Learning in Riemannian orbifolds is motivated by existing machine learning algorithms that directly operate on finite combinatorial structures such as point patterns, trees, and graphs. These methods, however, lack statistical justification. This contribution derives consistency results for learning problems in structured domains and thereby generalizes learning in vector spaces and manifolds.
Institute of Scientific and Technical Information of China (English)
Yan ZHANG
2007-01-01
When a target manifold is complete with a bounded curvature, we prove that there exists a unique global solution which satisfies the Euler-lagrange equation of (τ)(φ,(φ))=fR1+1{(|dφ|2+}dtdx for the given Cauchy data.
Stochastic Properties of the Laplacian on Riemannian Submersions
Brandão, M Cristiane
2011-01-01
Based on ideas of Pigolla and Setti \\cite{PS} we prove that immersed submanifolds with bounded mean curvature of Cartan-Hadamard manifolds are Feller. We also consider Riemannian submersions $\\pi \\colon M \\to N$ with compact minimal fibers, and based on various criteria for parabolicity and stochastic completeness, see \\cite{Grygor'yan}, we prove that $M$ is Feller, parabolic or stochastically complete if and only if the base $N$ is Feller, parabolic or stochastically complete respectively.
$(k,s)$-positivity and vanishing theorems for compact Kahler manifolds
Yang, Qi-Lin
2010-01-01
We study the $(k,s)$-positivity for holomorphic vector bundles on compact complex manifolds. $(0,s)$-positivity is exactly the Demailly $s$-positivity and a $(k,1)$-positive line bundle is just a $k$-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for $(k,s)$-positive vector bundles are proved and the vanishing theorems for $k$-ample vector bundles on projective algebraic manifolds are generalized to $k$-positive vector bundles on compact K\\"ahler manifolds.
Institute of Scientific and Technical Information of China (English)
刘元; 吴小俊
2015-01-01
In image recognition applications, Riemannian manifold learning algorithms can not eliminate the redundant information in images effectively. Therefore, an image recognition algorithm based on Log-Gabor wavelet and Riemannian manifold learning is presented. Firstly, images are processed by the Log-Gabor filter to obtain high-dimensional Log-Gabor image features. Then, the Riemannian manifold learning algorithm is used to reduce the dimensionality of the image features. Research shows that the integration of Log-Gabor wavelet and Riemannian manifold learning is in accord with the process of human visual perception. The proposed algorithm has better robustness to illumination and angle variation of the image. Experimental results on several standard databases indicate the effectiveness of the proposed algorithm.%在图像识别的研究中,黎曼流形学习不能有效消除图像中的冗余信息。基于上述原因,文中提出基于 Log-Gabor 滤波与黎曼流形学习的图像识别算法。首先使用 Log-Gabor 滤波器处理图像,获得维数较高的 Log-Gabor 图像特征,然后使用黎曼流形学习降维图像特征。研究表明,Log-Gabor 滤波与黎曼流形学习的融合算法符合人类视觉感知的过程。文中算法对于光照、角度变化具有较好的鲁棒性,在多个标准数据库上的仿真实验验证文中算法的有效性。
Manifold statistics for essential matrices
Dubbelman, G.; Dorst, L.; Pijls, H.; Fitzgibbon, A.; et al.,
2012-01-01
Riemannian geometry allows for the generalization of statistics designed for Euclidean vector spaces to Riemannian manifolds. It has recently gained popularity within computer vision as many relevant parameter spaces have such a Riemannian manifold structure. Approaches which exploit this have been
Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds
2001-01-01
The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. The proof is based on an elementary derivation of sharp Fredholm theorems for self-adjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds.
3-manifolds with(out) metrics of nonpositive curvature
Leeb, B
1994-01-01
In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly empty, boundary of zero Euler characteristic admits metrics of nonpositive curvature if the boundary is non-empty or if at least one atoroidal component occurs in its canonical topological decomposition. Our arguments are based on Thurstons Hyperbolisation Theorem. We give examples of closed graph-manifolds with linear gluing graph and arbitrarily many Seifert components which do not admit metrics of nonpositive curvature.
On stable compact minimal submanifolds
Torralbo, Francisco
2010-01-01
Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable compact minimal surfaces of the product of a 2-sphere and any Riemann surface are the complex ones.
A Class of Homogeneous Einstein Manifolds
Institute of Scientific and Technical Information of China (English)
Yifang KANG; Ke LIANG
2006-01-01
A Riemannian manifold (M,g) is called Einstein manifold if its Ricci tensor satisfies r=c·g for some constant c. General existence results are hard to obtain,e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.
Berezin-Toeplitz Quantization for Compact Kähler Manifolds. A Review of Results
Directory of Open Access Journals (Sweden)
Martin Schlichenmaier
2010-01-01
Full Text Available This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz deformation quantization for compact quantizable Kähler manifolds. The basic objects, concepts, and results are given. This concerns the correct semiclassical limit behaviour of the operator quantization, the unique Berezin-Toeplitz deformation quantization (star product, covariant and contravariant Berezin symbols, and Berezin transform. Other related objects and constructions are also discussed.
On the Mean Curvature of Semi-Riemannian Graphs in Semi-Riemannian Warped Products
Indian Academy of Sciences (India)
Zonglao Zhang
2012-08-01
We investigate the mean curvature of semi-Riemannian graphs in the semi-Riemannian warped product $M× f\\mathbb{R}_$, where is a semi-Riemannian manifold, $\\mathbb{R}_$ is the real line $\\mathbb{R}$ with metric $ dt^2( =± 1)$, and $f:M→ \\mathbb{R}^+$ is the warping function. We obtain an integral formula for mean curvature and some results dealing with estimates of mean curvature, among these results is a Heinz–Chern type inequality.
Eigenvalue pinching on spinc manifolds
Roos, Saskia
2017-02-01
We derive various pinching results for small Dirac eigenvalues using the classification of spinc and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spinc manifolds which involves a general study on convergence of Riemannian manifolds with a principal S1-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S1-bundle on spinc manifolds with Killing spinors.
CURVATURE COMPUTATIONS OF 2-MANIFOLDS IN IRk
Institute of Scientific and Technical Information of China (English)
Guo-liang Xu; Chandrajit L. Bajaj
2003-01-01
In this paper, we provide simple and explicit formulas for computing Riemannian cur-vatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IRk with k ≥ 3.
Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold
Institute of Scientific and Technical Information of China (English)
LIAO Shan-tao
2006-01-01
Let Mn be an n-dimensional compact C∞-differentiable manifold,n≥2,and let S be a C1 -differential system on Mn.The system induces a one-parameter C1 transformation group φt(-∞＜t＜∞) over Mn and,thus,naturally induces a one-parameter transformation group of the tangent bundle of Mn.The aim of this paper,in essence,is to study certain ergodic properties of this latter transformation group.Among various results established in the paper,we mention here only the following,which might describe quite well the nature of our study.(A) Let M be the set of regular points in Mn of the differential system S.With respect to a given C∞ Riemannian metric of Mn,we consider the bundle (ζ)# of all (n-2) spheres Qxn-2,x∈M,where Qxn-2 for each x consists of all unit tangent vectors of Mn orthogonal to the trajectory through x.Then,the differential system S gives rise naturally to a one-parameter transformation group Ψt#(-∞＜t＜∞) of (ζ)#.For an l-frame α=(u1,u2,…,ul) of Mn at a point x in M,1 ≥l≥n-1,each ui being in (ζ)#,we shall denote the volume of the parallelotope in the tangent space of Mn at x with edges u1,u2,…,ut by v(α),and let η*α(t)=v(Ψ#t (u1),Ψ#t(u2),…,Ψ#t (ul)).This is a continuous real function of t.Let I*+(α)=-limT→∞1/T∫T0η*α(t)dt,I*-(α)=-limT→-∞1/T∫T0η*α(t)dt.α is said to be positively linearly independent of the mean if I+*(α)＞0.Similarly,α is said to be negatively linearly independent of the mean if I_ *(a)＞0.A point x of M is said to possess positive generic index κ=κ+*(x) if,at x,there is a κ-frame α= (u1,u2,…,Uκ),ui ∈(ζ)#,of Mn having the property of being positively linearly independent in the mean,but at x,every l-frame β=(v1,v2,…,vl),vi ∈(ζ)#,of Mn with l＞κ does not have the same property.Similarly,we define the negative generic index κ_*(x) of x.For a nonempty closed subset F of Mn consisting of regular points of S,invariant under φt(-∞＜t＜∞),let the (positive and
Branson, T P; Vasilevich, D V
1998-01-01
Let M be a compact Riemannian manifold with smooth boundary. We study the vacuum expectation value of an operator Q by studying Tr Qe^{-tD}, where D is an operator of Laplace type on M, and where Q is a second order operator with scalar leading symbol; we impose Dirichlet or modified Neumann boundary conditions.
Comparison theorems in Riemannian geometry
Cheeger, Jeff
2008-01-01
The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. The first five chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov's theorem-the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter of Morse theory is followed by one on the injectivity radius. Chapters 6-9 deal with many of the most re
Institute of Scientific and Technical Information of China (English)
范宏; 邹翎; 王凯; 李柏林; 高业文
2011-01-01
Purpose To improve the precision of fiber tracking andto better understandthe normal brain function. Materials andMethods Based on the streamline tracking technology, this algorithm introduced Riemannian metrics and inverse distance weights tointerpolatethe diffusion tensor of current points. Results Using this new method, the fibers tracked from in vivo data were smoother and longer than those obtained using the conventional interpolation methods. Conclusion Riemannian manifold inverse distance interpolation is an effective fiber tracking technology.%目的 提高纤维追踪的精度,有助于理解正常脑功能.材料与方法在流线追踪的基础上,引入黎曼测度和反距离权重,从而完成对当前点的张量值插值.结果 新插值方法追踪出的纤维比传统插值方法的更长,更平滑.结论 该算法是一种有效的纤维追踪技术.
Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry
Eldering, Jaap
2012-01-01
We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.
Geometric control theory and sub-Riemannian geometry
Boscain, Ugo; Gauthier, Jean-Paul; Sarychev, Andrey; Sigalotti, Mario
2014-01-01
This volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume.
A parametric design of compact exhaust manifold junction in heavy duty diesel engine using CFD
Directory of Open Access Journals (Sweden)
Naeimi Hessamedin
2011-01-01
Full Text Available Nowadays, computational fluid dynamics codes (CFD are prevalently used to simulate the gas dynamics in many fluid piping systems such as steam and gas turbines, inlet and exhaust in internal combustion engines. In this paper, a CFD software is used to obtain the total energy losses in adiabatic compressible flow at compact exhaust manifold junction. A steady state onedimensional adiabatic compressible flow with friction model has been applied to subtract the straight pipe friction losses from the total energy losses. The total pressure loss coefficient has been related to the extrapolated Mach number in the common branch and to the mass flow rate ratio between branches at different flow configurations, in both combining and dividing flows. The study indicate that the numerical results were generally in good agreement with those of experimental data from the literature and will be applied as a boundary condition in one-dimensional global simulation models of fluid systems in which these components are present.
Cone fields and topological sampling in manifolds with bounded curvature
Turner, Katharine
2011-01-01
Often noisy point clouds are given as an approximation of a particular compact set of interest. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of {\\mu}-critical points in an annular region. Since an offset of a set deformation retracts to the set itself provided that there are no critical points of the distance function nearby, we can use this theorem to show when the offset of a point cloud is homotopy equivalent to the set it is sampled from. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature. In the process, we prove stability theorems for {\\mu}-critical points when the ambient space is a manifold.
Statistics on Lie groups: A need to go beyond the pseudo-Riemannian framework
Miolane, Nina; Pennec, Xavier
2015-01-01
Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group G is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall [1, 2, 3, 4] followed by others [5, 6, 7, 8, 9]. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is compatible with the group structure, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group G. The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space.
Indian Academy of Sciences (India)
Aydin Gezer
2009-06-01
The purpose of the present article is to investigate some relations between the Lie algebra of the infinitesimal fibre-preserving conformal transformations of the tangent bundle of a Riemannian manifold with respect to the synectic lift of the metric tensor and the Lie algebra of infinitesimal projective transformations of the Riemannian manifold itself.
Shiota, Masahiro
1987-01-01
A Nash manifold denotes a real manifold furnished with algebraic structure, following a theorem of Nash that a compact differentiable manifold can be imbedded in a Euclidean space so that the image is precisely such a manifold. This book, in which almost all results are very recent or unpublished, is an account of the theory of Nash manifolds, whose properties are clearer and more regular than those of differentiable or PL manifolds. Basic to the theory is an algebraic analogue of Whitney's Approximation Theorem. This theorem induces a "finiteness" of Nash manifold structures and differences between Nash and differentiable manifolds. The point of view of the author is topological. However the proofs also require results and techniques from other domains so elementary knowledge of commutative algebra, several complex variables, differential topology, PL topology and real singularities is required of the reader. The book is addressed to graduate students and researchers in differential topology and real algebra...
Semiclassical Asymptotics on Manifolds with Boundary
Koldan, Nilufer; Shubin, Mikhail
2008-01-01
We discuss semiclassical asymptotics for the eigenvalues of the Witten Laplacian for compact manifolds with boundary in the presence of a general Riemannian metric. To this end, we modify and use the variational method suggested by Kordyukov, Mathai and Shubin (2005), with a more extended use of quadratic forms instead of the operators. We also utilize some important ideas and technical elements from Helffer and Nier (2006), who were the first to supply a complete proof of the full semi-classical asymptotic expansions for the eigenvalues with fixed numbers.
The constraint equations for the Einstein-scalar field system on compact manifolds
Choquet-Bruhat, Y; Pollack, D; Choquet-Bruhat, Yvonne; Isenberg, James; Pollack, Daniel
2006-01-01
We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For many of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed sca...
The constraint equations for the Einstein-scalar field system on compact manifolds
Energy Technology Data Exchange (ETDEWEB)
Choquet-Bruhat, Yvonne [University of Paris VI, 4 place jussieu, 75005, Paris (France); Isenberg, James [Department of Mathematics, University of Oregon, Eugene, Oregon 97403-5203 (United States); Pollack, Daniel [Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350 (United States)
2007-02-21
We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For many of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases.
Dynamical Casimir Effect in a small compact manifold for the Maxwell vacuum
Zhitnitsky, Ariel R
2015-01-01
We study novel type of contributions to the partition function of the Maxwell system defined on a small compact manifold ${\\mathbb{M}}$ such as torus. These new terms can not be described in terms of the physical propagating photons with two transverse polarizations. Rather, these novel contributions emerge as a result of tunnelling events when transitions occur between topologically different but physically identical vacuum winding states. These new terms give an extra contribution to the Casimir pressure, yet to be measured. We argue that if the same system is considered in the background of a small external time-dependent magnetic field, than there will be emission of photons from the vacuum, similar to the Dynamical Casimir Effect (DCE) when real particles are radiated from the vacuum due to the time-dependent boundary conditions. The difference with conventional DCE is that the dynamics of the vacuum in our system is not related to the fluctuations of the conventional degrees of freedom, the virtual phot...
Institute of Scientific and Technical Information of China (English)
杨飞; 沈婧芳
2011-01-01
本文研究了黎曼流形上一类一般的曲率流问题.利用Perelman在Ricci流下导出体积单调性的方法,在初始流形完备非紧的情况下,获得了这类曲率流的一个单调性的体积公式,推广了Reto Müller在紧致情形的结果.%We consider a kind of general geometric flow problem. By utilizing Perelman's method under the Ricci flow, when the initial manifold is complete and noncompact, we obtain compact.
Simple Riemannian surfaces are scattering rigid
Wen, Haomin
2015-01-01
Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results about lens rigidity but very little is known for scattering rigidity. We will discuss the subtle difference between these two types of rigidities and prove that they are equiva...
Riemannian geometry in an orthogonal frame
Cartan, Elie Joseph
2001-01-01
Foreword by S S Chern. In 1926-27, Cartan gave a series of lectures in which he introduced exterior forms at the very beginning and used extensively orthogonal frames throughout to investigate the geometry of Riemannian manifolds. In this course he solved a series of problems in Euclidean and non-Euclidean spaces, as well as a series of variational problems on geodesics. In 1960, Sergei P Finikov translated from French into Russian his notes of these Cartan's lectures and published them as a book entitled Riemannian Geometry in an Orthogonal Frame. This book has many innovations, such as the n
Institute of Scientific and Technical Information of China (English)
许达允; 全哲勇; 金光植
2014-01-01
In Riemannian manifold,we defined a semi-symmetric proj ective conformal connection and consid-ered its properties.In particular cases,this connection reduces to several connections:semi-symmetric proj ec-tive connection,semi-symmetric conformal connection,symmetric proj ective conformal connection,proj ective connection,conformal connection and Levi-Civita connection.We also found forms of a semi-symmetric pro-j ective conformal connection satisfying the Schur’s theorem.And we considered necessary and sufficient condi-tion that a Riemannian manifold with a semi-symmetric proj ective conformal connection be a Riemannian mani-fold with constant curvature.%在黎曼流形上定义了一个半对称射影共形联络，并研究了其性质，同时指出这种联络在特殊情形下可成半对称射影联络、半对称共形联络、对称射影共形联络、射影联络、共形联络以及 Levi-Civita联络。在此基础上提出了几种能够满足 Schur定理的半对称射影共形联络的形式，并证明半对称射影共形联络的黎曼流形是常曲率黎曼流形的充分必要条件。
On Kähler–Norden Manifolds-Erratum
Indian Academy of Sciences (India)
M Iscan; A A Salimov
2009-02-01
This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of Riemannian curvature tensors and curvature scalars of Kähler–Norden manifolds using the theory of Tachibana operators is presented.
Diffusion Harmonics and Dual Geometry on Carnot Manifolds
Constantin, Sarah
The "curse of dimensionality" motivates the importance of techniques for computing low-dimensional approximations of high-dimensional data. It is often necessary to use nonlinear techniques to recover a low-dimensional manifold embedded via a nonlinear map in a high-dimensional space; this family of techniques is referred to as "manifold learning." The accuracy of manifold-learning-based approximations is founded on asymptotic results that assume the data is drawn from a low-dimensional Riemannian manifold. However, in natural datasets, this assumption is often overly restrictive. In the first part of this thesis we examine a more general class of manifolds known as Carnot manifolds, a type of sub-Riemannian manifold that governs natural phenomena such as chemical kinetics and configuration spaces of jointed objects. We find that diffusion maps can be generalized to Carnot manifolds and that the projection onto diffusion harmonics gives an almost isometric embedding; as a side effect, the diffusion distance is a computationally fast estimate for the shortest distance between two points on a Carnot manifold. We apply this theory to biochemical network data and observe that the chemical kinetics of the EGFR network are governed by a Carnot, but not Riemannian, manifold. In the second part of this thesis we examine the Heisenberg group, a classical example of a Carnot manifold. We obtain a representation-theoretic proof that the eigenfunctions of the sub-Laplacian on SU(2) approach the eigenfunctions of the sub-Laplacian on the Heisenberg group, in the limit as the radius of the sphere becomes large, in analogy with the limiting relationship between the Fourier series on the circle and the Fourier transform on the line. This result also illustrates how projecting onto the sub-Laplacian eigenfunctions of a non-compact Carnot manifold can be locally approximated by projecting onto the sub-Laplacian eigenfunctions of a tangent compact Carnot manifold. In the third part
Matrix models, 4D black holes and topological strings on non-compact Calabi-Yau manifolds
Danielsson, Ulf H.; Olsson, Martin E.; Vonk, Marcel
2004-11-01
We study the relation between c = 1 matrix models at self-dual radii and topological strings on non-compact Calabi-Yau manifolds. Particularly the special case of the deformed matrix model is investigated in detail. Using recent results on the equivalence of the partition function of topological strings and that of four dimensional BPS black holes, we are able to calculate the entropy of the black holes, using matrix models. In particular, we show how to deal with the divergences that arise as a result of the non-compactness of the Calabi-Yau. The main result is that the entropy of the black hole at zero temperature coincides with the canonical free energy of the matrix model, up to a proportionality constant given by the self-dual temperature of the matrix model.
The antipodal sets of compact symmetric spaces
National Research Council Canada - National Science Library
Liu, Xingda; Deng, Shaoqiang
2014-01-01
We study the antipodal set of a point in a compact Riemannian symmetric space. It turns out that we can give an explicit description of the antipodal set of a point in any connected simply connected compact Riemannian symmetric space...
Fedosov and Riemannian supermanifolds
Asorey, M
2008-01-01
Generalizations of symplectic and metric structures for supermanifolds are analyzed. Two types of structures are possible according to the even/odd character of the corresponding quadratic tensors. In the even case one has a very rich set of geometric structures: even symplectic supermanifolds (or, equivalently, supermanifolds with non-degenerate Poisson structures), even Fedosov supermanifolds and even Riemannian supermanifolds. The existence of relations among those structures is analyzed in some details. In the odd case, we show that odd Riemannian and Fedosov supermanifolds are characterized by a scalar curvature tensor. However, odd Riemannian supermanifolds can only have constant curvature.
Energy Technology Data Exchange (ETDEWEB)
Choquet-Bruhat, Yvonne [Universite Paris 6, 4 Place Jussieu, 75005 Paris (France)
2004-02-07
We give a general survey of the solution of the Einstein constraints by the conformal method on n-dimensional compact manifolds. We prove some new results about solutions with low regularity (solutions in H{sub 2} when n = 3) and solutions with unscaled sources.
An index formula for perturbed Dirac operators on Lie manifolds
Carvalho, Catarina
2011-01-01
We give an index formula for a class of Dirac operators coupled with unbounded potentials. More precisely, we study operators of the form P := D+ V, where D is a Dirac operators and V is an unbounded potential at infinity on a possibly non-compact manifold M_0. We assume that M_0 is a Lie manifold with compactification denoted M. Examples of Lie manifolds are provided by asymptotically Euclidean or asymptotically hyperbolic spaces. The potential V is required to be invertible outside a compact set K and V^{-1} extends to a smooth function on M\\K that vanishes on all faces of M in a controlled way. Using tools from analysis on non-compact Riemannian manifolds, we show that the computation of the index of P reduces to the computation of the index of an elliptic pseudodifferential operator of order zero on M_0 that is a multiplication operator at infinity. The index formula for P can then be obtained from earlier results. The proof also yields similar index formulas for Callias-type pseudodifferential operators ...
Riemannian geometry of fluctuation theory: An introduction
Velazquez, Luisberis
2016-05-01
Fluctuation geometry was recently proposed as a counterpart approach of Riemannian geometry of inference theory (information geometry), which describes the geometric features of the statistical manifold M of random events that are described by a family of continuous distributions dpξ(x|θ). This theory states a connection among geometry notions and statistical properties: separation distance as a measure of relative probabilities, curvature as a measure about the existence of irreducible statistical correlations, among others. In statistical mechanics, fluctuation geometry arises as the mathematical apparatus of a Riemannian extension of Einstein fluctuation theory, which is also closely related to Ruppeiner geometry of thermodynamics. Moreover, the curvature tensor allows to express some asymptotic formulae that account for the system fluctuating behavior beyond the gaussian approximation, while curvature scalar appears as a second-order correction of Legendre transformation between thermodynamic potentials.
Brane world in Non-Riemannian Geometry
Maier, Rodrigo; 10.1103/PhysRevD.83.064019
2012-01-01
We carefully investigate the modified Einstein's field equation in a four dimensional (3-brane) arbitrary manifold embedded in a five dimensional Non-Riemannian bulk spacetime with a noncompact extra dimension. In this context the Israel-Darmois matching conditions are extended assuming that the torsion in the bulk is continuous. The discontinuity in the torsion first derivatives are related to the matter distribution through the field equation. In addition, we develop a model that describes a flat FLRW model embedded in a 5-dimensional de Sitter or Anti de Sitter, where a 5-dimensional cosmological constant emerges from the torsion.
Eisenhart, L P
1927-01-01
The use of the differential geometry of a Riemannian space in the mathematical formulation of physical theories led to important developments in the geometry of such spaces. The concept of parallelism of vectors, as introduced by Levi-Civita, gave rise to a theory of the affine properties of a Riemannian space. Covariant differentiation, as developed by Christoffel and Ricci, is a fundamental process in this theory. Various writers, notably Eddington, Einstein and Weyl, in their efforts to formulate a combined theory of gravitation and electromagnetism, proposed a simultaneous generalization o
Extended Riemannian Geometry I: Local Double Field Theory
Deser, Andreas
2016-01-01
We present an extended version of Riemannian geometry suitable for the description of current formulations of double field theory (DFT). This framework is based on graded manifolds and it yields extended notions of symmetries, dynamical data and constraints. In special cases, we recover general relativity with and without 1-, 2- and 3-form gauge potentials as well as DFT. We believe that our extended Riemannian geometry helps to clarify the role of various constructions in DFT. For example, it leads to a covariant form of the strong section condition. Furthermore, it should provide a useful step towards global and coordinate invariant descriptions of T- and U-duality invariant field theories.
Geometry of almost-product Lorentzian manifolds and relativistic observer
Borowiec, Andrzej
2013-01-01
The notion of relativistic observer is confronted with Naveira's classification of (pseudo-)Riemannian almost-product structures on space-time manifolds. Some physical properties and their geometrical counterparts are shortly discussed.
Riemannian geometric approach to human arm dynamics, movement optimization, and invariance.
Biess, Armin; Flash, Tamar; Liebermann, Dario G
2011-03-01
We present a generally covariant formulation of human arm dynamics and optimization principles in Riemannian configuration space. We extend the one-parameter family of mean-squared-derivative (MSD) cost functionals from Euclidean to Riemannian space, and we show that they are mathematically identical to the corresponding dynamic costs when formulated in a Riemannian space equipped with the kinetic energy metric. In particular, we derive the equivalence of the minimum-jerk and minimum-torque change models in this metric space. Solutions of the one-parameter family of MSD variational problems in Riemannian space are given by (reparameterized) geodesic paths, which correspond to movements with least muscular effort. Finally, movement invariants are derived from symmetries of the Riemannian manifold. We argue that the geometrical structure imposed on the arm's configuration space may provide insights into the emerging properties of the movements generated by the motor system.
Akbulut, Selman
2010-01-01
It is known that every compact Stein 4-manifolds can be embedded into a simply connected, minimal, closed, symplectic 4-manifold. Using this property, we give simple constructions of various cork structures of 4-manifolds. We also give an example of infinitely many disjoint embeddings of a fixed cork into a non-compact 4-manifold which produce infinitely many exotic smooth structures (recall that [7] gives examples arbitrarily many disjoint imbeddings of different corks in a closed manifold inducing mutually different exotic structures). Furthermore, here we construct arbitrary many simply connected compact codimention zero submanifolds of S^4 which are mutually homeomorphic but not diffeomorphic.
Natural Diagonal Riemannian Almost Product and Para-Hermitian Cotangent Bundles
Druta-Romaniuc, Simona-Luiza
2011-01-01
We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem for the natural diagonal (almost) para-K\\"ahlerian structures on the total spaces of the cotangent bundle.
Post-Riemannian approach for the symplectic and elliptic geometries of gravity
Energy Technology Data Exchange (ETDEWEB)
Cartas-Fuentevilla, R; Solano-Altamirano, J M [Instituto de Fisica, Universidad Autonoma de Puebla, Apartado postal J-48 72570 Puebla Pue. (Mexico); Enriquez-Silverio, P, E-mail: rcartas@sirio.ifuap.buap.mx [Facultad de Ciencias FIsico Matematicas, Universidad Autonoma de Puebla, Apartado postal 1152, 72001 Puebla Pue. (Mexico)
2011-05-13
Considering a post-Riemannian approach for manifolds where metric and connection are not necessarily compatible, the symplectic geometry of the covariant phase space of gravity is considered; the symplectic structure associated allows us to study the Poincare charges and the weak-field approach at the asymptotic region, specifically the effects of torsional contributions through metric deformations. Additionally, the elliptic geometry of the moduli space of gravitational instantons is considered along the same lines, which provides the grounds for the construction of the partition function of the theory and new invariants for smooth four-manifolds from a post-Riemannian theoretic point of view.
A dense G-delta set of Riemannian metrics without the finite blocking property
Gerber, Marlies
2010-01-01
A pair of points (x,y) in a Riemannian manifold (M,g) is said to have the finite blocking property if there is a finite set P contained in M\\{x,y} such that every geodesic segment from x to y passes through a point of P. We show that for every closed C-infinity manifold M of dimension at least two and every pair (x,y) in M x M, there exists a dense G-delta set of C-infinity Riemannian metrics on M such that (x,y) fails to have the finite blocking property for every g in that set.
The Superspinorial Field Theory in Riemannian Coordinates
Derbenev, Yaroslav
2016-01-01
The Superspinorial Dual-covariant Field Theory (SSFT) developed in papers [1, 2] is treated in terms of Riemannian coordinates (RC) [7, 8] in space of the N dimensions unified manifold (UM). Metric tensor of UM (grand metric, GM) is built on the split metric matrices (SM) [1] which are a proportion of the Cartan's affinors (an extended analog of Dirac's matrices) of his Theory of Spinors [3] as explicated in [2]. Transition to RC based on consideration of geodesics is described. A principal property of an orthogonal RC frame (ORC) utilized in the present paper is constancy of the rotation matrix A of the Riemannian space of UM, while transformation matrix B of the dual superspinorial state vector field (DSV) varies together with Cartan's affinors according to the dynamical law of SSFT derived in [2]. The spinorial genesis of notion of the orthogonality as aspect of irreducible SSFT is pointed out in the present paper. The main outcome of resorting to an orthogonal RC frame (ORC) is explication of the conforma...
Riemannian Geometry: Definitions, Pictures, and Results
Marsh, Adam
2014-01-01
A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. As much material as possible is developed for manifolds with connection (omitting a metric) to make clear which aspects can be readily generalized to gauge theories. The presentation in most cases does not assume a coordinate frame or zero torsion, and the coordinate-free, tensor, and Cartan formalisms are developed in parallel.
Determining the first order perturbation of a polyharmonic operator on admissible manifolds
Assylbekov, Yernat M.; Yang, Yang
2017-01-01
We consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator L g , X , q, with X being a W 1 , ∞ vector field and q being an L∞ function on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Dirichlet-to-Neumann map determines X and q uniquely. The method is based on the construction of complex geometrical optics solutions using the Carleman estimate for the Laplace-Beltrami operator due to Dos Santos Ferreira, Kenig, Salo and Uhlmann. Notice that the corresponding uniqueness result does not hold for the first order perturbation of the Laplace-Beltrami operator.
An introduction to differential manifolds
Lafontaine, Jacques
2015-01-01
This book is an introduction to differential manifolds. It gives solid preliminaries for more advanced topics: Riemannian manifolds, differential topology, Lie theory. It presupposes little background: the reader is only expected to master basic differential calculus, and a little point-set topology. The book covers the main topics of differential geometry: manifolds, tangent space, vector fields, differential forms, Lie groups, and a few more sophisticated topics such as de Rham cohomology, degree theory and the Gauss-Bonnet theorem for surfaces. Its ambition is to give solid foundations. In particular, the introduction of “abstract” notions such as manifolds or differential forms is motivated via questions and examples from mathematics or theoretical physics. More than 150 exercises, some of them easy and classical, some others more sophisticated, will help the beginner as well as the more expert reader. Solutions are provided for most of them. The book should be of interest to various readers: undergra...
Local Schrodinger flow into Kahler manifolds
Institute of Scientific and Technical Information of China (English)
DlNG; Weiyue(
2001-01-01
［1］Ding, W. Y. , Wang, Y. D. , Schrodinger flows of maps into symplectic manifolds, Science in China, Ser. A, 1998, 41(7): 746.［2］Landau, L. D., Lifshitz, E. M., On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z.Sowj., 1935, 8: 153; reproduced in Collected Papers of L. D. Landau, New York: Pergaman Press, 1965, 101－114.［3］Faddeev, L., Takhtajan, L. A. , Hamiltonian Methods in the Theory of Solitons, Berlin-Heidelberg-New York: Springer-Verlag, 1987.［4］Nakamura, K., Sasada, T., Soliton and wave trains in ferromagnets, Phys. Lett. A, 1974, 48: 321.［5］Zhou, Y. , Guo, B. , Tan, S. , Existence and uniqueness of smooth solution for system of ferromagnetic chain, Science in China, Ser. A, 1991, 34(3): 257.［6］Pang, P. , Wang, H. , Wang, Y. D. , Schrodinger flow of maps into Kahler manifolds, Asian J. of Math. , in press.［7］Wang, H. , Wang, Y. D. , Global inhomogeneous Schrodinger flow, Int. J. Math., 2000, 11: 1079.［8］Pang, P., Wang, H., Wang, Y. D., Local existence for inhomogeneous Schrodinger flow of maps into Kahler manifolds,Acta Math. Sinica, English Series, 2000, 16: 487.［9］Temg, C. L., Uhlenbeck, K., Schrodinger flows on Grassmannians, in Integrable Systems, Geometry and Topology,Somervi11e, MA: International Press, in press.［10］Chang, N., Shatah, J., Uhlenbeck, K., Schrodinger maps, Commun. Pure Appl. Math., 2000, 53: 157.［11］Wang, Y. D., Ferromagnetic chain equation from a closed Riemannian manifold into S2, Int. J. Math., 1995, 6: 93.［12］Wang, Y. D., Heisenberg chain systems from compact manifolds into S2, J. Math. Phys., 1998, 39(1): 363.［13］Sulem, P., Sulem, C., Bardos, C., On the continuous limit for a system of classical spins, Commun. Math. Phys., 1986,107: 431.［14］Aubin, T., Nonlinear Analysis on Manifolds, Monge-Ampère Equations, Berlin-Heidelberg-New York: Springer-Verlag,1982.［15］Eells, J. , Lemaire, L. , Another report on harmonic maps, Bull. London
On the Stability of the $L^p$-Norm of the Riemannian Curvature Tensor
Indian Academy of Sciences (India)
Soma Maity
2014-08-01
We consider the Riemannian functional $\\mathcal{R}_p(g)=\\int_M|R(g)|^p dv_g$ defined on the space of Riemannian metrics with unit volume on a closed smooth manifold where $R(g)$ and $dv_g$ denote the corresponding Riemannian curvature tensor and volume form and $p\\in (0,∞)$. First we prove that the Riemannian metrics with non-zero constant sectional curvature are strictly stable for $\\mathcal{R}_p$ for certain values of . Then we conclude that they are strict local minimizers for $\\mathcal{R}_p$ for those values of . Finally generalizing this result we prove that product of space forms of same type and dimension are strict local minimizer for $\\mathcal{R}_p$ for certain values of .
How to Find the Holonomy Algebra of a Lorentzian Manifold
Galaev, Anton S.
2015-02-01
Manifolds with exceptional holonomy play an important role in string theory, supergravity and M-theory. It is explained how one can find the holonomy algebra of an arbitrary Riemannian or Lorentzian manifold. Using the de Rham and Wu decompositions, this problem is reduced to the case of locally indecomposable manifolds. In the case of locally indecomposable Riemannian manifolds, it is known that the holonomy algebra can be found from the analysis of special geometric structures on the manifold. If the holonomy algebra of a locally indecomposable Lorentzian manifold ( M, g) of dimension n is different from , then it is contained in the similitude algebra . There are four types of such holonomy algebras. Criterion to find the type of is given, and special geometric structures corresponding to each type are described. To each there is a canonically associated subalgebra . An algorithm to find is provided.
Resonant oscillations in ${\\alpha}^{2}$-dynamos on a closed, twisted Riemannian 2D flux tubes
de Andrade, Garcia
2009-01-01
Chicone et al [CMP (1995)] have shown that, kinematic fast dynamos in diffusive media, could exist only on a closed, 2D Riemannian manifold of constant negative curvature. This report, shows that their result cannot be extended to oscillatory ${\\alpha}^{2}$-dynamos, when there are resonance modes, between toroidal and poloidal frequencies of twisted magnetic flux tubes. Thus, dynamo action can be supported in regions, where Riemannian curvature is positive. For turbulent dynamos, this seems physically reasonable, since recently, [Shukurov et al PRE (2008)] have obtained a Moebius flow strip in sodium liquid, torus Perm dynamo where curvature is also connected to the magnetic fields via diffusion. This could be done, by adjusting the corresponding frequencies till they achieved resonance. Actually 2D torus, is a manifold of zero mean curvature, where regions of positive and negative curvatures exist. It is shown that, Riemannian solitonic surface, endowed with a steady ${\\alpha}^{2}$-dynamo from magnetic filam...
Progress in the Theory of Singular Riemannian Foliations
Alexandrino, Marcos M; Toeben, Dirk
2012-01-01
A singular foliation is called a singular Riemannian foliation (SRF) if every geodesic that is perpendicular to one leaf is perpendicular to every leaf it meets. A typical example is the partition of a complete Riemannian manifold into orbits of an isometric action. In this survey, we provide an introduction to the theory of SRFs, leading from the foundations to recent developments in research on this subject. Sketches of proofs are included and useful techniques are emphasized. We study the local structure of SRFs in general and under curvature conditions. We review the solution of the Palais-Terng problem on integrability of the horizontal distribution. Important special classes of SRFs, like polar and variationally complete foliations and their connections, are treated. A characterisation of SRFs whose leaf space is an orbifold is given. Moreover, desingularizations of SRFs are studied and applications, e.g., to Molino's conjecture, are presented.
Xi, Yakun; Zhang, Cheng
2016-07-01
We show that one can obtain improved L 4 geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge's strategy in (Improved critical eigenfunction estimates on manifolds of nonpositive curvature, Preprint). We first combine the improved L 2 restriction estimate of Blair and Sogge (Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint) and the classical improved {L^∞} estimate of Bérard to obtain an improved weak-type L 4 restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger (Math Res Lett 18(4):767-781, 2011). This estimate improves the L 4 restriction estimate of Burq et al. (Duke Math J 138:445-486, 2007) and Hu (Forum Math 6:1021-1052, 2009) by a power of {(log logλ)^{-1}} . Moreover, in the case of compact hyperbolic surfaces, we obtain further improvements in terms of {(logλ)^{-1}} by applying the ideas from (Chen and Sogge, Commun Math Phys 329(3):435-459, 2014) and (Blair and Sogge, Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint). We are able to compute various constants that appeared in (Chen and Sogge, Commun Math Phys 329(3):435-459, 2014) explicitly, by proving detailed oscillatory integral estimates and lifting calculations to the universal cover H^2.
Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$
2012-01-01
We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$. The sub-Riemannian distance makes $H^1$ a metric space and consenquently with a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transverse to the tangent space of S. If all points of S have this property, we prove a Gauss-Bonnet formula and for compact surfaces (which are topologically a torus) we obtain $\\int_S ...
A rigidity theorem for complete noncompact Bach-flat manifolds
Chu, Yawei
2011-02-01
Let (M4,g) be a four-dimensional complete noncompact Bach-flat Riemannian manifold with positive Yamabe constant. In this paper, we show that (M4,g) has a constant curvature if it has a nonnegative constant scalar curvature and sufficiently small L2-norm of trace-free Riemannian curvature tensor. Moreover, we get a gap theorem for (M4,g) with positive scalar curvature.
Energy Technology Data Exchange (ETDEWEB)
Caballero, Magdalena; Rubio, Rafael M [Departamento de Matematicas, Campus de Rabanales, Universidad de Cordoba, 14071 Cordoba (Spain); Romero, Alfonso, E-mail: magdalena.caballero@uco.es, E-mail: aromero@ugr.es, E-mail: rmrubio@uco.es [Departamento de Geometria y Topologia, Universidad de Granada, 18071 Granada (Spain)
2011-07-21
A new technique to study spacelike hypersurfaces of constant mean curvature in a spacetime which admits a timelike gradient conformal vector field is introduced. As an application, the leaves of the natural spacelike foliation of such spacetimes are characterized in some relevant cases. The global structure of this class of spacetimes is analyzed and the relation with its well-known subfamily of generalized Robertson-Walker spacetimes is exposed in detail. Moreover, some known uniqueness results for compact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes are widely extended. Finally, and as a consequence, several Calabi-Bernstein problems are solved obtaining all the entire solutions on a compact Riemannian manifold to the constant mean curvature spacelike hypersurface equation, under natural geometric assumptions.
Riemannian-geometric entropy for measuring network complexity
Franzosi, Roberto; Felice, Domenico; Mancini, Stefano; Pettini, Marco
2016-06-01
A central issue in the science of complex systems is the quantitative characterization of complexity. In the present work we address this issue by resorting to information geometry. Actually we propose a constructive way to associate with a—in principle, any—network a differentiable object (a Riemannian manifold) whose volume is used to define the entropy. The effectiveness of the latter in measuring network complexity is successfully proved through its capability of detecting a classical phase transition occurring in both random graphs and scale-free networks, as well as of characterizing small exponential random graphs, configuration models, and real networks.
Eignets for function approximation on manifolds
Mhaskar, H N
2009-01-01
Let $\\XX$ be a compact, smooth, connected, Riemannian manifold without boundary, $G:\\XX\\times\\XX\\to \\RR$ be a kernel. Analogous to a radial basis function network, an eignet is an expression of the form $\\sum_{j=1}^M a_jG(\\circ,y_j)$, where $a_j\\in\\RR$, $y_j\\in\\XX$, $1\\le j\\le M$. We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in $L^p(\\mu;\\XX)$ for a general class of measures $\\mu$ and kernels $G$. Our algorithm yields linear operators. Using the minimal separation amongst the centers $y_j$ as the cost of approximation, we give modulus of smoothness estimates for the degree of approximation by our eignets, and show by means of a converse theorem that these are the best possible for every \\emph{individual function}. We also give estimates on the coefficients $a_j$ in terms of the norm of the eignet. Finally, we demonstrate that if any sequence of eignets satisfies the optimal estimates for the degree of approximation of a smooth function, measured in ter...
Flux formulation of DFT on group manifolds and generalized Scherk-Schwarz compactifications
Energy Technology Data Exchange (ETDEWEB)
Bosque, Pascal du [Max-Planck-Institut für Physik,Föhringer Ring 6, 80805 München (Germany); Arnold-Sommerfeld-Center für Theoretische Physik,Fakultät für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany); Hassler, Falk [University of North Carolina, Department of Physics and Astronomy,Phillips Hall, CB #3255, 120 E. Cameron Ave., Chapel Hill, NC 27599-3255 (United States); City University of New York, The Graduate Center,365 Fifth Avenue, New York, NY 10016 (United States); Columbia University, Department of Physics,Pupin Hall, 550 West 120th St., New York, NY 10027 (United States); Lüst, Dieter [Max-Planck-Institut für Physik,Föhringer Ring 6, 80805 München (Germany); Arnold-Sommerfeld-Center für Theoretische Physik,Fakultät für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany)
2016-02-04
A flux formulation of Double Field Theory on group manifold is derived and applied to study generalized Scherk-Schwarz compactifications, which give rise to a bosonic subsector of half-maximal, electrically gauged supergravities. In contrast to the flux formulation of original DFT, the covariant fluxes split into a fluctuation and a background part. The latter is connected to a 2D-dimensional, pseudo Riemannian manifold, which is isomorphic to a Lie group embedded into O(D,D). All fields and parameters of generalized diffeomorphisms are supported on this manifold, whose metric is spanned by the background vielbein E{sub A}{sup I}∈ GL(2D). This vielbein takes the role of the twist in conventional generalized Scherk-Schwarz compactifications. By doing so, it solves the long standing problem of constructing an appropriate twist for each solution of the embedding tensor. Using the geometric structure, absent in original DFT, E{sub A}{sup I} is identified with the left invariant Maurer-Cartan form on the group manifold, in the same way as it is done in geometric Scherk-Schwarz reductions. We show in detail how the Maurer-Cartan form for semisimple and solvable Lie groups is constructed starting from the Lie algebra. For all compact embeddings in O(3,3), we calculate E{sub A}{sup I}.
Warped product submanifolds of Lorentzian paracosymplectic manifolds
Perkta\\cs, Selcen Yüksel; Kele\\cs, Sad\\ik
2011-01-01
In this paper we study the warped product submanifolds of a Lorentzian paracosymplectic manifold and obtain some nonexistence results. We show that a warped product semi-invariant submanifold in the form {$M=M_{T}\\times_{f}M_{\\bot}$} of Lorentzian paracosymplectic manifold such that the characteristic vector field is normal to $M$ is an usual Riemannian product manifold where totally geodesic and totally umbilical submanifolds of warped product are invariant and anti-invariant, respectively. We prove that the distributions involved in the definition of a warped product semi-invariant submanifold are always integrable. A necessary and sufficient condition for a semi-invariant submanifold of a Lorentzian paracosymplectic manifold to be warped product semi-invariant submanifold is obtained. We also investigate the existence and nonexistence of warped product semi-slant and warped product anti-slant submanifolds in a Lorentzian paracosymplectic manifold.
Geometry and physics of pseudodifferential operators on manifolds
Esposito, Giampiero; Napolitano, George M.
2016-09-01
A review is made of the basic tools used in mathematics to define a calculus for pseudodifferential operators on Riemannian manifolds endowed with a connection: existence theorem for the function that generalizes the phase; analogue of Taylor's theorem; torsion and curvature terms in the symbolic calculus; the two kinds of derivative acting on smooth sections of the cotangent bundle of the Riemannian manifold; the concept of symbol as an equivalence class. Physical motivations and applications are then outlined, with emphasis on Green functions of quantum field theory and Parker's evaluation of Hawking radiation.
SL(2;R)/U(1) supercoset and elliptic genera of Non-compact Calabi-Yau Manifolds
Eguchi, T
2004-01-01
We first discuss the relationship between the SL(2;)/U(1) supercoset and = 2 Liouville theory and make a precise correspondence between their representations. We shall show that the discrete unitary representations of SL(2;)/U(1) theory correspond exactly to those massless representations of = 2 Liouville theory which are closed under modular transformations and studied in our previous work [18]. It is known that toroidal partition functions of SL(2;)/U(1) theory (2D Black Hole) contain two parts, continuous and discrete representations. The contribution of continuous representations is proportional to the space-time volume and is divergent in the infinite-volume limit while the part of discrete representations is volume-independent. In order to see clearly the contribution of discrete representations we consider elliptic genus which projects out the contributions of continuous representations: making use of the SL(2;)/U(1), we compute elliptic genera for various non-compact space-times such as the conifold, ...
HOLOMORPHIC MANIFOLDS ON LOCALLY CONVEX SPACES
Institute of Scientific and Technical Information of China (English)
Tsoy-Wo Ma
2005-01-01
Based on locally compact perturbations of the identity map similar to the Fredholm structures on real Banach manifolds, complex manifolds with inverse mapping theorem as part of the defintion are proposed. Standard topics including holomorphic maps, morphisms, derivatives, tangent bundles, product manifolds and submanifolds are presented. Although this framework is elementary, it lays the necessary foundation for all subsequent developments.
Dark Energy and Dark Matter From Hidden Symmetry of Gravity Model with a Non-Riemannian Volume Form
Guendelman, Eduardo; Pacheva, Svetlana
2015-01-01
We show that dark energy and dark matter can be described simultaneously by ordinary Einstein gravity interacting with a single scalar field provided the scalar field Lagrangian couples in a symmetric fashion to two different spacetime volume-forms (covariant integration measure densities) on the spacetime manifold - one standard Riemannian given by the square-root of the determinant of the pertinent Riemannian metric and another non-Riemannian volume-form independent of the Riemannian metric, defined in terms of an auxiliary antisymmetric tensor gauge field of maximal rank. Integration of the equations of motion of the latter auxiliary gauge field produce an a priori arbitrary integration constant that plays the role of a dynamically generated cosmological constant or dark energy. Moreover, the above modified scalar field action turns out to possess a hidden Noether symmetry whose associated conserved current describes a pressureless "dust" fluid which we can identify with the dark matter completely decouple...
Riemannian computing in computer vision
Srivastava, Anuj
2016-01-01
This book presents a comprehensive treatise on Riemannian geometric computations and related statistical inferences in several computer vision problems. This edited volume includes chapter contributions from leading figures in the field of computer vision who are applying Riemannian geometric approaches in problems such as face recognition, activity recognition, object detection, biomedical image analysis, and structure-from-motion. Some of the mathematical entities that necessitate a geometric analysis include rotation matrices (e.g. in modeling camera motion), stick figures (e.g. for activity recognition), subspace comparisons (e.g. in face recognition), symmetric positive-definite matrices (e.g. in diffusion tensor imaging), and function-spaces (e.g. in studying shapes of closed contours). · Illustrates Riemannian computing theory on applications in computer vision, machine learning, and robotics · Emphasis on algorithmic advances that will allow re-application in other...
The Hodge theory of projective manifolds
de Cataldo, Mark Andrea
2007-01-01
This book is a written-up and expanded version of eight lectures on the Hodge theory of projective manifolds. It assumes very little background and aims at describing how the theory becomes progressively richer and more beautiful as one specializes from Riemannian, to Kähler, to complex projective manifolds. Though the proof of the Hodge Theorem is omitted, its consequences - topological, geometrical and algebraic - are discussed at some length. The special properties of complex projective manifolds constitute an important body of knowledge and readers are guided through it with the help of selected exercises. Despite starting with very few prerequisites, the concluding chapter works out, in the meaningful special case of surfaces, the proof of a special property of maps between complex projective manifolds, which was discovered only quite recently.
D\\'{e}formations isospectrales non compactes et th\\'{e}orie quantique des champs
Gayral, V
2005-01-01
The aim of this thesis is to study the isopectral deformations from the point of view of Alain Connes' noncommutative geometry. This class of quantum spaces constituts a curved space generalisation of Moyal planes and noncommutative tori. First of all, we look at the construction of non-unital spectral triples, for which we propose modified axioms. We then check that Moyal planes fit into this axiomatic framework, and give the keypoints for the construction of non-unital spectral triples from generic non-compact isospectral deformations. To this end, numerous analytical tools on non-compact Riemannian manifolds are developped. Thanks to Dixmier traces computations, we show that their spectral and classical dimensions coincide. In a second time, we study certain features of quantum fields theory on curved isospectral deformations, with a particular view on the ultraviolet infrared mixing phenomenon. We show its intrinsic nature for all such quantum spaces (compacts or not, periodic or not deformations), and we...
Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry
Eldering, J
2012-01-01
We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. The $C^{k,\\alpha}$-smoothness result is optimal with respect to the spectral gap condition involved. The core of the persistence proof is based on the Perron method. In the process we derive new results on noncompact submanifolds in bounded geometry: a uniform tubular neighborhood theorem and uniform smooth approximation of a submanifold. The submanifolds considered are assumed to be uniformly $C^k$ bounded in an appropriate sense.
Pro jective vector fields on Finsler manifolds
Institute of Scientific and Technical Information of China (English)
TIAN Huang-jia
2014-01-01
In this paper, we give the equation that characterizes projective vector fields on a Finsler manifold by the local coordinate. Moreover, we obtain a feature of the projective fields on the compact Finsler manifold with non-positive flag curvature and the non-existence of projective vector fields on the compact Finsler manifold with negative flag curvature. Furthermore, we deduce some expectable, but non-trivial relationships between geometric vector fields such as projective, affine, conformal, homothetic and Killing vector fields on a Finsler manifold.
Stability of Strongly Gauduchon Manifolds under Modifications
Popovici, Dan
2010-01-01
In our previous works on deformation limits of projective and Moishezon manifolds, we introduced and made crucial use of the notion of strongly Gauduchon metrics as a reinforcement of the earlier notion of Gauduchon metrics. Using direct and inverse images of closed positive currents of type $(1, \\, 1)$ and regularisation, we now show that compact complex manifolds carrying strongly Gauduchon metrics are stable under modifications. This stability property, known to fail for compact K\\"ahler manifolds, mirrors the modification stability of balanced manifolds proved by Alessandrini and Bassanelli.
Estimation of Radio Interferometer Beam Shapes Using Riemannian Optimization
Yatawatta, Sarod
2012-01-01
The knowledge of receiver beam shapes is essential for accurate radio interferometric imaging. Traditionally, this information is obtained by holographic techniques or by numerical simulation. However, such methods are not feasible for an observation with time varying beams, such as the beams produced by a phased array radio interferometer. We propose the use of the observed data itself for the estimation of the beam shapes. We use the directional gains obtained along multiple sources across the sky for the construction of a time varying beam model. The construction of this model is an ill posed non linear optimization problem. Therefore, we propose to use Riemannian optimization, where we consider the constraints imposed as a manifold. We compare the performance of the proposed approach with traditional unconstrained optimization and give results to show the superiority of the proposed approach.
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
Casetti, L; Pettini, M; Casetti, Lapo; Clementi, Cecilia; Pettini, Marco
1996-01-01
This paper deals with the problem of analytically computing the largest Lyapunov exponent for many degrees of freedom Hamiltonian systems. This aim is succesfully reached within a theoretical framework that makes use of a geometrization of newtonian dynamics in the language of Riemannian geometry. A new point of view about the origin of chaos in these systems is obtained independently of homoclinic intersections. Chaos is here related to curvature fluctuations of the manifolds whose geodesics are natural motions and is described by means of Jacobi equation for geodesic spread. Under general conditions ane effective stability equation is derived; an analytic formula for the growth-rate of its solutions is worked out and applied to the Fermi-Pasta-Ulam beta model and to a chain of coupled rotators. An excellent agreement is found the theoretical prediction and the values of the Lyapunov exponent obtained by numerical simulations for both models.
Singular reduction of generalized complex manifolds
Goldberg, Timothy E
2010-01-01
In this paper, we develop the analogue of Sjamaar and Lerman's singular reduction of Hamiltonian symplectic manifolds in the context of Hamiltonian generalized complex manifolds (in the sense of Lin and Tolman). Specifically, we prove that if a compact Lie group acts on a generalized complex manifold in a Hamiltonian fashion, then the partition of the global quotient by orbit types induces a partition of the Lin-Tolman quotient into generalized complex manifolds. This result holds also for reduction of Hamiltonian generalized Kaehler manifolds.
The infinity(x-Laplace equation in Riemannian vector fields
Directory of Open Access Journals (Sweden)
Thomas Bieske
2015-06-01
Full Text Available We employ Riemannian jets which are adapted to the Riemannian geometry to obtain the existence-uniqueness of viscosity solutions to the infinity(x-Laplace equation in Riemannian vector fields. Due to the differences between Euclidean jets and Riemannian jets, the Euclidean method of proof is not valid in this environment.
Degenerate pseudo-Riemannian metrics
Hervik, Sigbjorn; Yamamoto, Kei
2014-01-01
In this paper we study pseudo-Riemannian spaces with a degenerate curvature structure i.e. there exists a continuous family of metrics having identical polynomial curvature invariants. We approach this problem by utilising an idea coming from invariant theory. This involves the existence of a boost, the existence of this boost is assumed to extend to a neighbourhood. This approach proves to be very fruitful: It produces a class of metrics containing all known examples of degenerate metrics. To date, only Kundt and Walker metrics have been given, however, our study gives a plethora of examples showing that degenerate metrics extend beyond the Kundt and Walker examples. The approach also gives a useful criterion for a metric to be degenerate. Specifically, we use this to study the subclass of VSI and CSI metrics (i.e., spaces where polynomial curvature invariants are all vanishing or constants, respectively).
Manifolds, tensors and, forms an introduction for mathematicians and physicists
Renteln, Paul
2014-01-01
Providing a succinct yet comprehensive treatment of the essentials of modern differential geometry and topology, this book's clear prose and informal style make it accessible to advanced undergraduate and graduate students in mathematics and the physical sciences. The text covers the basics of multilinear algebra, differentiation and integration on manifolds, Lie groups and Lie algebras, homotopy and de Rham cohomology, homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and degree theory. It also features over 250 detailed exercises, and a variety of applications revealing fundamental connections to classical mechanics, electromagnetism (including circuit theory), general relativity and gauge theory. Solutions to the problems are available for instructors at www.cambridge.org/9781107042193.
Unconstrained steepest descent method for multicriteria optimization on Riemmanian manifolds
Bento, G C; Oliveira, P R
2010-01-01
In this paper we present a steepest descent method with Armijo's rule for multicriteria optimization in the Riemannian context. The well definedness of the sequence generated by the method is guaranteed. Under mild assumptions on the multicriteria function, we prove that each accumulation point (if they exist) satisfies first-order necessary conditions for Pareto optimality. Moreover, assuming quasi-convexity of the multicriteria function and non-negative curvature of the Riemannian manifold, we prove full convergence of the sequence to a Pareto critical.
Duality constructions from quantum state manifolds
Kriel, J N; Scholtz, F G
2015-01-01
The formalism of quantum state space geometry on manifolds of generalised coherent states is proposed as a natural setting for the construction of geometric dual descriptions of non-relativistic quantum systems. These state manifolds are equipped with natural Riemannian and symplectic structures derived from the Hilbert space inner product. This approach allows for the systematic construction of geometries which reflect the dynamical symmetries of the quantum system under consideration. We analyse here in detail the two dimensional case and demonstrate how existing results in the AdS_2/CFT_1 context can be understood within this framework. We show how the radial/bulk coordinate emerges as an energy scale associated with a regularisation procedure and find that, under quite general conditions, these state manifolds are asymptotically anti-de Sitter solutions of a class of classical dilaton gravity models. For the model of conformal quantum mechanics proposed by de Alfaro et. al. the corresponding state manifol...
Dissertation: Geodesics of Random Riemannian Metrics
LaGatta, Tom
2011-01-01
We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on $\\mathbb R^d$. We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one. In differential geometry, geodesics are curves which locally minimize length. They need not do so globally: consider great circles on a sphere. For lattice models of FPP, there are many open questions related to minimizing geodesics; similarly, it is interesting from a geometric perspective when geodesics are globally minimizing. In the present study, we show that for any fixed st...
Du, Jia; Qiu, Anqi
2011-01-01
In this paper, we propose a novel large deformation diffeomorphic registration algorithm to align high angular resolution diffusion images (HARDI) characterized by orientation distribution functions (ODFs). Our proposed algorithm seeks an optimal diffeomorphism of large deformation between two ODF fields in a spatial volume domain and at the same time, locally reorients an ODF in a manner such that it remains consistent with the surrounding anatomical structure. To this end, we first review the Riemannian manifold of ODFs. We then define the reorientation of an ODF when an affine transformation is applied and subsequently, define the diffeomorphic group action to be applied on the ODF based on this reorientation. We incorporate the Riemannian metric of ODFs for quantifying the similarity of two HARDI images into a variational problem defined under the large deformation diffeomorphic metric mapping (LDDMM) framework. We finally derive the gradient of the cost function in both Riemannian spaces of diffeomorphis...
Directory of Open Access Journals (Sweden)
Pąk Karol
2015-02-01
Full Text Available Let us recall that a topological space M is a topological manifold if M is second-countable Hausdorff and locally Euclidean, i.e. each point has a neighborhood that is homeomorphic to an open ball of E n for some n. However, if we would like to consider a topological manifold with a boundary, we have to extend this definition. Therefore, we introduce here the concept of a locally Euclidean space that covers both cases (with and without a boundary, i.e. where each point has a neighborhood that is homeomorphic to a closed ball of En for some n.
Kosinski, Antoni A
2007-01-01
The concepts of differential topology form the center of many mathematical disciplines such as differential geometry and Lie group theory. Differential Manifolds presents to advanced undergraduates and graduate students the systematic study of the topological structure of smooth manifolds. Author Antoni A. Kosinski, Professor Emeritus of Mathematics at Rutgers University, offers an accessible approach to both the h-cobordism theorem and the classification of differential structures on spheres.""How useful it is,"" noted the Bulletin of the American Mathematical Society, ""to have a single, sho
Manifold learning on brain functional networks in aging.
Qiu, Anqi; Lee, Annie; Tan, Mingzhen; Chung, Moo K
2015-02-01
We propose a new analysis framework to utilize the full information of brain functional networks for computing the mean of a set of brain functional networks and embedding brain functional networks into a low-dimensional space in which traditional regression and classification analyses can be easily employed. For this, we first represent the brain functional network by a symmetric positive matrix computed using sparse inverse covariance estimation. We then impose a Log-Euclidean Riemannian manifold structure on brain functional networks whose norm gives a convenient and practical way to define a mean. Finally, based on the fact that the computation of linear operations can be done in the tangent space of this Riemannian manifold, we adopt Locally Linear Embedding (LLE) to the Log-Euclidean Riemannian manifold space in order to embed the brain functional networks into a low-dimensional space. We show that the integration of the Log-Euclidean manifold with LLE provides more efficient and succinct representation of the functional network and facilitates regression analysis, such as ridge regression, on the brain functional network to more accurately predict age when compared to that of the Euclidean space of functional networks with LLE. Interestingly, using the Log-Euclidean analysis framework, we demonstrate the integration and segregation of cortical-subcortical networks as well as among the salience, executive, and emotional networks across lifespan.
Aytuna, Aydin
2011-01-01
An open Riemann surface is called parabolic in case every bounded subharmonic function on it reduces to a constant. Several authors introduced seemingly different analogs of this notion for Stein manifolds of arbitrary dimension. In the first part of this note we compile these notions of parabolicity and give some immediate relations among them. In section 3 we relate some of these notions to the linear topological type of the Fr\\'echet space of analytic functions on the given manifold. In sections 4 and 5 we look at some examples and show, for example, that the complement of the zero set of a Weierstrass polynomial possesses a continuous plurisubharmonic exhaustion function that is maximal off a compact subset.
Nearly pseudo-Kähler manifolds and related special holonomies
Schäfer, Lars
2017-01-01
Developing and providing an overview of recent results on nearly Kähler geometry on pseudo-Riemannian manifolds, this monograph emphasizes the differences with the classical Riemannian geometry setting. The focal objects of the text are related to special holonomy and Killing spinors and have applications in high energy physics, such as supergravity and string theory. Before starting into the field, a self-contained introduction to the subject is given, aimed at students with a solid background in differential geometry. The book will therefore be accessible to masters and Ph.D. students who are beginning work on nearly Kähler geometry in pseudo-Riemannian signature, and also to non-experts interested in gaining an overview of the subject. Moreover, a number of results and techniques are provided which will be helpful for differential geometers as well as for high energy physicists interested in the mathematical background of the geometric objects they need.
Hashing on nonlinear manifolds.
Shen, Fumin; Shen, Chunhua; Shi, Qinfeng; van den Hengel, Anton; Tang, Zhenmin; Shen, Heng Tao
2015-06-01
Learning-based hashing methods have attracted considerable attention due to their ability to greatly increase the scale at which existing algorithms may operate. Most of these methods are designed to generate binary codes preserving the Euclidean similarity in the original space. Manifold learning techniques, in contrast, are better able to model the intrinsic structure embedded in the original high-dimensional data. The complexities of these models, and the problems with out-of-sample data, have previously rendered them unsuitable for application to large-scale embedding, however. In this paper, how to learn compact binary embeddings on their intrinsic manifolds is considered. In order to address the above-mentioned difficulties, an efficient, inductive solution to the out-of-sample data problem, and a process by which nonparametric manifold learning may be used as the basis of a hashing method are proposed. The proposed approach thus allows the development of a range of new hashing techniques exploiting the flexibility of the wide variety of manifold learning approaches available. It is particularly shown that hashing on the basis of t-distributed stochastic neighbor embedding outperforms state-of-the-art hashing methods on large-scale benchmark data sets, and is very effective for image classification with very short code lengths. It is shown that the proposed framework can be further improved, for example, by minimizing the quantization error with learned orthogonal rotations without much computation overhead. In addition, a supervised inductive manifold hashing framework is developed by incorporating the label information, which is shown to greatly advance the semantic retrieval performance.
Xie, Xiaofeng; Yu, Zhu Liang; Lu, Haiping; Gu, Zhenghui; Li, Yuanqing
2016-07-07
In motor imagery brain-computer interfaces (BCIs), the symmetric positive-definite (SPD) covariance matrices of electroencephalogram (EEG) signals carry important discriminative information. In this paper, we intend to classify motor imagery EEG signals by exploiting the fact that the space of SPD matrices endowed with Riemannian distance is a highdimensional Riemannian manifold. To alleviate the overfitting and heavy computation problems associated with conventional classification methods on high-dimensional manifold, we propose a framework for intrinsic sub-manifold learning from a high-dimensional Riemannian manifold. Considering a special case of SPD space, a simple yet efficient bilinear sub-manifold learning (BSML) algorithm is derived to learn the intrinsic submanifold by identifying a bilinear mapping that maximizes the preservation of the local geometry and global structure of the original manifold. Two BSML-based classification algorithms are further proposed to classify the data on a learned intrinsic sub-manifold. Experimental evaluation of the classification of EEG revealed that the BSML method extracts the intrinsic submanifold approximately 5 faster and with higher classification accuracy compared with competing algorithms. The BSML also exhibited strong robustness against a small training dataset, which often occurs in BCI studies.
Holomorphic Cartan geometries, Calabi--Yau manifolds and rational curves
Biswas, Indranil; 10.1016/j.difgeo.2009.09.003
2010-01-01
We prove that if a Calabi--Yau manifold $M$ admits a holomorphic Cartan geometry, then $M$ is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact K\\"ahler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projective manifolds.
Local topology in deformation spaces of hyperbolic 3-manifolds
Brock, Jeffrey F; Canary, Richard D; Minsky, Yair N
2009-01-01
We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3-manifolds and Bers slices.
Foucault pendulum and sub-Riemannian geometry
Anzaldo-Meneses, A.; Monroy-Pérez, F.
2010-08-01
The well known Foucault nonsymmetrical pendulum is studied as a problem of sub-Riemannian geometry on nilpotent Lie groups. It is shown that in a rotating frame a sub-Riemannian structure can be naturally introduced. For small oscillations, three dimensional horizontal trajectories are computed and displayed in detail. The fiber bundle structure is explicitly shown. The underlying Lie structure is described together with the corresponding holonomy group, which turns out to be given by the center of the Heisenberg group. Other related physical problems that can be treated in a similar way are also mentioned.
Non-Riemannian geometrical optics in QED
Garcia de Andrade, L C
2003-01-01
A non-minimal photon-torsion axial coupling in the quantum electrodynamics (QED) framework is considered. The geometrical optics in Riemannian-Cartan spacetime is considering and a plane wave expansion of the electromagnetic vector potential is considered leading to a set of the equations for the ray congruence. Since we are interested mainly on the torsion effects in this first report we just consider the Riemann-flat case composed of the Minkowskian spacetime with torsion. It is also shown that in torsionic de Sitter background the vacuum polarisation does alter the propagation of individual photons, an effect which is absent in Riemannian spaces.
Rapid Mixing of Geodesic Walks on Manifolds with Positive Curvature
Mangoubi, Oren; Smith, Aaron
2016-01-01
We introduce a Markov chain for sampling from the uniform distribution on a Riemannian manifold $\\mathcal{M}$, which we call the $\\textit{geodesic walk}$. We prove that the mixing time of this walk on any manifold with positive sectional curvature $C_{x}(u,v)$ bounded both above and below by $0 < \\mathfrak{m}_{2} \\leq C_{x}(u,v) \\leq \\mathfrak{M}_2 < \\infty$ is $\\mathcal{O}^*\\left(\\frac{\\mathfrak{M}_2}{\\mathfrak{m}_2}\\right)$. In particular, this bound on the mixing time does not depend expli...
New spinor fields on Lorentzian 7-manifolds
Energy Technology Data Exchange (ETDEWEB)
Bonora, L. [International School for Advanced Studies (SISSA),Via Bonomea 265, 34136 Trieste (Italy); Rocha, Roldão da [Centro de Matemática, Computação e Cognição, Universidade Federal do ABC,Avenida dos Estados, 5001, Santo André (Brazil)
2016-01-21
This paper deals with the classification of spinor fields according to the bilinear covariants in 7 dimensions. The previously investigated Riemannian case is characterized by either one spinor field class, in the real case of Majorana spinors, or three non-trivial classes in the most general complex case. In this paper we show that by imposing appropriate conditions on spinor fields in 7d manifolds with Lorentzian metric, the formerly obtained obstructions for new classes of spinor fields can be circumvented. New spinor fields classes are then explicitly constructed. In particular, on 7-manifolds with asymptotically flat black hole background, these spinors can define a generalized current density which further defines a time Killing vector at the spatial infinity.
A study on neural learning on manifold foliations: the case of the Lie group SU(3).
Fiori, Simone
2008-04-01
Learning on differential manifolds may involve the optimization of a function of many parameters. In this letter, we deal with Riemannian-gradient-based optimization on a Lie group, namely, the group of unitary unimodular matrices SU(3). In this special case, subalgebras of the associated Lie algebra su(3) may be individuated by computing pair-wise Gell-Mann matrices commutators. Subalgebras generate subgroups of a Lie group, as well as manifold foliation. We show that the Riemannian gradient may be projected over tangent structures to foliation, giving rise to foliation gradients. Exponentiations of foliation gradients may be computed in closed forms, which closely resemble Rodriguez forms for the special orthogonal group SO(3). We thus compare optimization by Riemannian gradient and foliation gradients.
ON THE FUNDAMENTAL GROUP OF OPEN MANIFOLDS WITH NONNEGATIVE RICCI CURVATURE
Institute of Scientific and Technical Information of China (English)
XU SENLIN; WANG ZUOQIN; YANG FANGYUN
2003-01-01
The authors establish some uniform estimates for the distance to halfway points of minimalgeodesics in terms of the distantce to end points on some types of Riemannian manifolds, andthen prove some theorems about the finite generation of fundamental group of Riemannianmanifold with nonnegative Ricci curvature, which support the famous Milnor conjecture.
On the classical geometry of embedded manifolds in terms of Nambu brackets
Arnlind, Joakim; Huisken, Gerhard
2010-01-01
We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of a multi-linear algebraic structure on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.
An atomic decomposition of the Haj{\\l}asz Sobolev space $\\Mone$ on manifolds
Badr, Nadine
2009-01-01
Several possible notions of Hardy-Sobolev spaces on a Riemannian manifold with a doubling measure are considered. Under the assumption of a Poincar\\'e inequality, the space $\\Mone$, defined by Haj{\\l}asz, is identified with a Hardy-Sobolev space defined in terms of atoms. Decomposition results are proved for both the homogeneous and the nonhomogeneous spaces.
Estimates and Nonexistence of Solutions of the Scalar Curvature Equation on Noncompact Manifolds
Indian Academy of Sciences (India)
Zhang Zonglao
2005-08-01
This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.
Contour Propagation With Riemannian Elasticity Regularization
DEFF Research Database (Denmark)
Bjerre, Troels; Hansen, Mads Fogtmann; Sapru, W.;
2011-01-01
the planning CT onto the rescans and correcting to reflect actual anatomical changes. For deformable registration, a free-form, multi-level, B-spline deformation model with Riemannian elasticity, penalizing non-rigid local deformations, and volumetric changes, was used. Regularization parameters was defined...
Geometric inequalities in sub-Riemannian groups
Montefalcone, Francescopaolo
2012-01-01
Let G be a sub-Riemannian k-step Carnot group of homogeneous dimension Q. In this paper, we shall prove several geometric inequalities concerning smooth hypersurfaces (i.e. codimension one submanifolds) immersed in G, endowed with the H-perimeter measure.
Parallel spinors on flat manifolds
Sadowski, Michał
2006-05-01
Let p(M) be the dimension of the vector space of parallel spinors on a closed spin manifold M. We prove that every finite group G is the holonomy group of a closed flat spin manifold M(G) such that p(M(G))>0. If the holonomy group Hol(M) of M is cyclic, then we give an explicit formula for p(M) another than that given in [R.J. Miatello, R.A. Podesta, The spectrum of twisted Dirac operators on compact flat manifolds, Trans. Am. Math. Soc., in press]. We answer the question when p(M)>0 if Hol(M) is a cyclic group of prime order or dimM≤4.
Mao, Shasha; Xiong, Lin; Jiao, Licheng; Feng, Tian; Yeung, Sai-Kit
2016-07-26
Riemannian optimization has been widely used to deal with the fixed low-rank matrix completion problem, and Riemannian metric is a crucial factor of obtaining the search direction in Riemannian optimization. This paper proposes a new Riemannian metric via simultaneously considering the Riemannian geometry structure and the scaling information, which is smoothly varying and invariant along the equivalence class. The proposed metric can make a tradeoff between the Riemannian geometry structure and the scaling information effectively. Essentially, it can be viewed as a generalization of some existing metrics. Based on the proposed Riemanian metric, we also design a Riemannian nonlinear conjugate gradient algorithm, which can efficiently solve the fixed low-rank matrix completion problem. By experimenting on the fixed low-rank matrix completion, collaborative filtering, and image and video recovery, it illustrates that the proposed method is superior to the state-of-the-art methods on the convergence efficiency and the numerical performance.
Duality constructions from quantum state manifolds
Kriel, J. N.; van Zyl, H. J. R.; Scholtz, F. G.
2015-11-01
The formalism of quantum state space geometry on manifolds of generalised coherent states is proposed as a natural setting for the construction of geometric dual descriptions of non-relativistic quantum systems. These state manifolds are equipped with natural Riemannian and symplectic structures derived from the Hilbert space inner product. This approach allows for the systematic construction of geometries which reflect the dynamical symmetries of the quantum system under consideration. We analyse here in detail the two dimensional case and demonstrate how existing results in the AdS 2 /CF T 1 context can be understood within this framework. We show how the radial/bulk coordinate emerges as an energy scale associated with a regularisation procedure and find that, under quite general conditions, these state manifolds are asymptotically anti-de Sitter solutions of a class of classical dilaton gravity models. For the model of conformal quantum mechanics proposed by de Alfaro et al. [1] the corresponding state manifold is seen to be exactly AdS 2 with a scalar curvature determined by the representation of the symmetry algebra. It is also shown that the dilaton field itself is given by the quantum mechanical expectation values of the dynamical symmetry generators and as a result exhibits dynamics equivalent to that of a conformal mechanical system.
A compactness theorem for surfaces with Bounded Integral Curvature
Debin, Clément
2016-01-01
We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $\\Sigma$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation of singularities is allowed.
Dark energy and dark matter from hidden symmetry of gravity model with a non-Riemannian volume form
Energy Technology Data Exchange (ETDEWEB)
Guendelman, Eduardo [Ben-Gurion University of the Negev, Department of Physics, Beersheba (Israel); Nissimov, Emil; Pacheva, Svetlana [Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, Sofia (Bulgaria)
2015-10-15
We show that dark energy and dark matter can be described simultaneously by ordinary Einstein gravity interacting with a single scalar field provided the scalar field Lagrangian couples in a symmetric fashion to two different spacetime volume forms (covariant integration measure densities) on the spacetime manifold - one standard Riemannian given by √(-g) (square root of the determinant of the pertinent Riemannian metric) and another non-Riemannian volume form independent of the Riemannian metric, defined in terms of an auxiliary antisymmetric tensor gauge field of maximal rank. Integration of the equations of motion of the latter auxiliary gauge field produce an a priori arbitrary integration constant that plays the role of a dynamically generated cosmological constant or dark energy. Moreover, the above modified scalar field action turns out to possess a hidden Noether symmetry whose associated conserved current describes a pressureless ''dust'' fluid which we can identify with the dark matter completely decoupled from the dark energy. The form of both the dark energy and dark matter that results from the above class of models is insensitive to the specific form of the scalar field Lagrangian. By adding an appropriate perturbation, which breaks the above hidden symmetry and along with this couples dark matter and dark energy, we also suggest a way to obtain growing dark energy in the present universe's epoch without evolution pathologies. (orig.)
Iwasawa nilpotency degree of non compact symmetric cosets in N-extended Supergravity
Cacciatori, Sergio Luigi; Ferrara, Sergio; Marrani, Alessio
2014-01-01
We analyze the polynomial part of the Iwasawa realization of the coset representative of non compact symmetric Riemannian spaces. We start by studying the role of Kostant's principal SU(2)_P subalgebra of simple Lie algebras, and how it determines the structure of the nilpotent subalgebras. This allows us to compute the maximal degree of the polynomials for all faithful representations of Lie algebras. In particular the metric coefficients are related to the scalar kinetic terms while the representation of electric and magnetic charges is related to the coupling of scalars to vector field strengths as they appear in the Lagrangian. We consider symmetric scalar manifolds in N-extended supergravity in various space-time dimensions, elucidating various relations with the underlying Jordan algebras and normed Hurwitz algebras. For magic supergravity theories, our results are consistent with the Tits-Satake projection of symmetric spaces and the nilpotency degree turns out to depend only on the space-time dimensio...
Deformations of extremal toric manifolds
Rollin, Yann
2012-01-01
Let $X$ be a compact toric extremal K\\"ahler manifold. Using the work of Sz\\'ekelyhidi, we provide a simple criterion on the fan describing $X$ to ensure the existence of complex deformations of $X$ that carry extremal metrics. As an example, we find new CSC metrics on 4-points blow-ups of $\\C\\P^1\\times\\C\\P^1$.
On Einstein, Hermitian 4-Manifolds
LeBrun, Claude
2010-01-01
Let (M,h) be a compact 4-dimensional Einstein manifold, and suppose that h is Hermitian with respect to some complex structure J on M. Then either (M,J,h) is Kaehler-Einstein, or else, up to rescaling and isometry, it is one of the following two exceptions: the Page metric on CP2 # (-CP2), or the Einstein metric on CP2 # 2 (-CP2) constructed in Chen-LeBrun-Weber.
Sub-Riemannian geometry and optimal transport
Rifford, Ludovic
2014-01-01
The book provides an introduction to sub-Riemannian geometry and optimal transport and presents some of the recent progress in these two fields. The text is completely self-contained: the linear discussion, containing all the proofs of the stated results, leads the reader step by step from the notion of distribution at the very beginning to the existence of optimal transport maps for Lipschitz sub-Riemannian structure. The combination of geometry presented from an analytic point of view and of optimal transport, makes the book interesting for a very large community. This set of notes grew from a series of lectures given by the author during a CIMPA school in Beirut, Lebanon.
Symmetries of sub-Riemannian surfaces
Malakhaltsev, Mikhail Armenovich
2009-01-01
Given a contact distribution $(\\Delta, )$ in $\\mathbf{R}^{3}$ the problem to determinate all symmetries of this sub-Riemannian surface with metric $$ was solved by Hughen \\cite{Hughen}, and completely by Montgomery \\cite{Montgomery}. Our goal is to obtain explicit formulae for this solution. We obtain explicit formulae for the functions which define symmetries in terms of a local coordinate system and explicit formulae for the invariants in terms of the dual frame and the structure functions.
Weyl-Euler-Lagrange Equations of Motion on Flat Manifold
Directory of Open Access Journals (Sweden)
Zeki Kasap
2015-01-01
Full Text Available This paper deals with Weyl-Euler-Lagrange equations of motion on flat manifold. It is well known that a Riemannian manifold is said to be flat if its curvature is everywhere zero. Furthermore, a flat manifold is one Euclidean space in terms of distances. Weyl introduced a metric with a conformal transformation for unified theory in 1918. Classical mechanics is one of the major subfields of mechanics. Also, one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange equations. In this study, partial differential equations have been obtained for movement of objects in space and solutions of these equations have been generated by using the symbolic Algebra software. Additionally, the improvements, obtained in this study, will be presented.
Rigidity of complete noncompact bach-flat n-manifolds
Chu, Yawei; Feng, Pinghua
2012-11-01
Let (Mn,g) be a complete noncompact Bach-flat n-manifold with the positive Yamabe constant and constant scalar curvature. Assume that the L2-norm of the trace-free Riemannian curvature tensor R∘m is finite. In this paper, we prove that (Mn,g) is a constant curvature space if the L-norm of R∘m is sufficiently small. Moreover, we get a gap theorem for (Mn,g) with positive scalar curvature. This can be viewed as a generalization of our earlier results of 4-dimensional Bach-flat manifolds with constant scalar curvature R≥0 [Y.W. Chu, A rigidity theorem for complete noncompact Bach-flat manifolds, J. Geom. Phys. 61 (2011) 516-521]. Furthermore, when n>9, we derive a rigidity result for R<0.
Total Variation Regularization for Functions with Values in a Manifold
Lellmann, Jan
2013-12-01
While total variation is among the most popular regularizers for variational problems, its extension to functions with values in a manifold is an open problem. In this paper, we propose the first algorithm to solve such problems which applies to arbitrary Riemannian manifolds. The key idea is to reformulate the variational problem as a multilabel optimization problem with an infinite number of labels. This leads to a hard optimization problem which can be approximately solved using convex relaxation techniques. The framework can be easily adapted to different manifolds including spheres and three-dimensional rotations, and allows to obtain accurate solutions even with a relatively coarse discretization. With numerous examples we demonstrate that the proposed framework can be applied to variational models that incorporate chromaticity values, normal fields, or camera trajectories. © 2013 IEEE.
Connections in sub-Riemannian geometry of parallelizable distributions
Youssef, Nabil L
2016-01-01
The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint of absolute parallelism geometry and sub-Riemannian geometry. Two remarkable linear connections have been constructed on a sub-Riemannian parallelizable distribution, namely, the Weitzenb\\"ock connection and the sub-Riemannian connection. The obtained results have been applied to two concrete examples: the spheres $S^3$ and $S^7$.
Two-phase Flow Distribution in Heat Exchanger Manifolds
Vist, Sivert
2004-01-01
The current study has investigated two-phase refrigerant flow distribution in heat exchange manifolds. Experimental data have been acquired in a heat exchanger test rig specially made for measurement of mass flow rate and gas and liquid distribution in the manifolds of compact heat exchangers. Twelve different manifold designs were used in the experiments, and CO2 and HFC-134a were used as refrigerants.
Some hyperbolic three-manifolds that bound geometrically
KOLPAKOV, Alexander; Martelli, Bruno; Tschantz, Steven
2015-01-01
A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension. We construct here infinitely many explicit examples in dimension $n=3$ using right-angled dodecahedra and $120$-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, fo...
Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds
Liu, Chiu-Chu Melissa; Sheshmani, Artan
2017-07-01
An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.
Generalized Einstein Tensor for a Weyl Manifold and Its Applications
Institute of Scientific and Technical Information of China (English)
Abdülkadir （O）ZDE（G）ER
2013-01-01
It is well known that the Einstein tensor G for a Piemannian manifold defined by Gβα =Rβα-1/2Rδα,Rβα =gβγRγα where Rγα and R are respectively the Ricci tensor and the scalar curvature of the manifold,plays an important part in Einstein's theory of gravitation as well as in proving some theorems in Riemannian geometry.In this work,we first obtain the generalized Einstein tensor for a Weyl manifold.Then,after studying some properties of generalized Einstein tensor,we prove that the conformed invariance of the generalized Einstein tensor implies the conformed invariance of the curvature tensor of the Weyl manifold and conversely.Moreover,we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogoned trajectories of which are geodesics.Finally,a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.
Renteln, Paul
2013-11-01
Preface; 1. Linear algebra; 2. Multilinear algebra; 3. Differentiation on manifolds; 4. Homotopy and de Rham cohomology; 5. Elementary homology theory; 6. Integration on manifolds; 7. Vector bundles; 8. Geometric manifolds; 9. The degree of a smooth map; Appendixes; References; Index.
Funar, L
1995-01-01
The aim of this note is to derive some invariants at infinity for open 3-manifolds in the framework of Topological Quantum Field Theories. These invariants may be used to test if an open manifold is simply connected at infinity as we done for Whitehead's manifold in case of the sl_{2}({\\bf C})-TQFT in level 4.
Cork twisting exotic Stein 4-manifolds
Akbulut, Selman
2011-01-01
From any 4-dimensional oriented handlebody X without 3- and 4-handles and with $b_2\\geq 1$, we construct arbitrary many compact Stein 4-manifolds which are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and intersection forms) coincide with those of X. We also discuss the induced contact structures on their boundaries. Furthermore, for any smooth 4-manifold pair (Z,Y) such that the complement $Z-\\textnormal{int}\\,Y$ is a handlebody without 3- and 4-handles and with $b_2\\geq 1$, we construct arbitrary many exotic embeddings of a compact 4-manifold Y' into Z, such that Y' has the same topological invariants as Y.
Piecewise linear manifolds: Einstein metrics and Ricci flows
Schrader, Robert
2016-05-01
This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear (p.l.) spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field. On a given set of p.l. spaces we define and discuss (normalized) Einstein flows. p.l. Einstein metrics are defined and examples are provided. Criteria for flows to approach Einstein metrics are formulated. Second variations of the total scalar curvature at a specific Einstein space are calculated. Dedicated to Ludwig Faddeev on the occasion of his 80th birthday.
Infinity Behavior of Bounded Subharmonic Functions on Ricci Non-negative Manifolds
Institute of Scientific and Technical Information of China (English)
Bao Qiang WU
2004-01-01
In this paper, we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that limr→∞r2/V(r) ∫B(r)△hdv = 0 if h is a bounded subharmonic function. If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity, under certain auxiliary conditions on the volume growth of M. In particular, our result applies to the case when the Riemannian manifold has maximum volume growth. We also derive a representation formula in our paper, from which one can easily derive Yau's Liouville theorem on bounded harmonic functions.
An investigation of embeddings for spherically symmetric spacetimes into Einstein manifolds
Indian Academy of Sciences (India)
Jothi Moodley; Gareth Amery
2011-09-01
Embeddings into higher dimensions are very important in the study of higherdimensional theories of our Universe and in high-energy physics. Theorems which have been developed recently guarantee the existence of embeddings of pseudo-Riemannian manifolds into Einstein spaces and more general pseudo-Riemannian spaces. These results provide a technique that can be used to determine solutions for such embeddings. Here we consider local isometric embeddings of four-dimensional spherically symmetric spacetimes into ﬁve-dimensional Einstein manifolds. Difﬁculties in solving the ﬁve-dimensional equations for given four-dimensional spaces motivate us to investigate embedded spaces that admit bulks of a speciﬁc type. We show that the general Schwarzschild–de Sitter spacetime and Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space of a particular form, and we discuss their ﬁve-dimensional solutions.
A global Torelli theorem for hyperkaehler manifolds (after Verbitsky)
Huybrechts, Daniel
2011-01-01
Compact hyperkaehler manifolds are higher-dimensional generalizations of K3 surfaces. The classical Global Torelli theorem for K3 surfaces, however, does not hold in higher dimensions. More precisely, a compact hyperkaehler manifold is in general not determined by its natural weight-two Hodge structure. The text gives an account of a recent theorem of M. Verbitsky, which can be regarded as a weaker version of the Global Torelli theorem phrased in terms of the injectivity of the period map on the connected components of the moduli space of marked manifolds.
The Ricci Curvature of Half-flat Manifolds
Ali, T; Ali, Tibra; Cleaver, Gerald B.
2007-01-01
We derive expressions for the Ricci curvature tensor and scalar in terms of intrinsic torsion classes of half-flat manifolds by exploiting the relationship between half-flat manifolds and non-compact $G_2$ holonomy manifolds. Our expressions are tested for Iwasawa and more general nilpotent manifolds. We also derive expressions, in the language of Calabi-Yau moduli spaces, for the torsion classes and the Ricci curvature of the \\emph{particular} half-flat manifolds that arise naturally via mirror symmetry in flux compactifications. Using these expressions we then derive a constraint on the K\\"ahler moduli space of type II string theory on these half-flat manifolds.
The Structure of some Classes of -Contact Manifolds
Indian Academy of Sciences (India)
Mukut Mani Tripathi; Mohit Kumar Dwivedi
2008-08-01
We study projective curvature tensor in -contact and Sasakian manifolds. We prove that (1) if a -contact manifold is quasi projectively flat then it is Einstein and (2) a -contact manifold is -projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a -contact manifold to be quasi projectively flat and -projectively flat are obtained. We also prove that for a (2+1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, -projectively flat and locally isometric to the unit sphere $S^{2n+1}(1)$ are equivalent. Finally, we prove that a compact -projectively flat -contact manifold with regular contact vector field is a principal $S^1$-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.
On the realizable topology of a manifold with attractors of geodesics
Fenille, Marcio Colombo
2017-01-01
We discuss the so-called realizable topology of a Riemannian manifold with attractors of geodesics, which we understand as its topological properties, mainly that related to its fundamental group, investigated from a viewpoint that may be considered realizable in a sense. In the special approach in which the manifold is understood as a model physical universe, we conclude that its realizable fundamental group is isomorphic to the classical fundamental group of its observable portion. For a universe of dimension at least three whose unobservable components are all contractible, this conclusion ensures the possibility to get real inferences about its classical fundamental group through observational methods.
Boots, Byron
2011-01-01
Recently, there has been much interest in spectral approaches to learning manifolds---so-called kernel eigenmap methods. These methods have had some successes, but their applicability is limited because they are not robust to noise. To address this limitation, we look at two-manifold problems, in which we simultaneously reconstruct two related manifolds, each representing a different view of the same data. By solving these interconnected learning problems together and allowing information to flow between them, two-manifold algorithms are able to succeed where a non-integrated approach would fail: each view allows us to suppress noise in the other, reducing bias in the same way that an instrumental variable allows us to remove bias in a {linear} dimensionality reduction problem. We propose a class of algorithms for two-manifold problems, based on spectral decomposition of cross-covariance operators in Hilbert space. Finally, we discuss situations where two-manifold problems are useful, and demonstrate that sol...
Riemannian means on special euclidean group and unipotent matrices group.
Duan, Xiaomin; Sun, Huafei; Peng, Linyu
2013-01-01
Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results.
DYNAMICS IN NEWTONIAN-RIEMANNIAN SPACE-TIME(Ⅳ)
Institute of Scientific and Technical Information of China (English)
张荣业
2001-01-01
Lagrangian mechanics in Newtonian-Riemannian space-time and relationship between Lagrangian mechanics and Newtonian mechanics, and between Lagrangian mechanics and Hamiltonian mechanics in N-R space-time are discussed.
Pseudo-Reimannian manifolds endowed with an almost para f-structure
Directory of Open Access Journals (Sweden)
Vladislav V. Goldberg
1985-01-01
Full Text Available Let M˜(U,Ω˜,η˜,ξ,g˜ be a pseudo-Riemannian manifold of signature (n+1,n. One defines on M˜ an almost cosymplectic para f-structure and proves that a manifold M˜ endowed with such a structure is ξ-Ricci flat and is foliated by minimal hypersurfaces normal to ξ, which are of Otsuki's type. Further one considers on M˜ a 2(n−1-dimensional involutive distribution P⊥ and a recurrent vector field V˜. It is proved that the maximal integral manifold M⊥ of P⊥ has V as the mean curvature vector (up to 1/2(n−1. If the complimentary orthogonal distribution P of P⊥ is also involutive, then the whole manifold M˜ is foliate. Different other properties regarding the vector field V˜ are discussed.
Borok, S.; Goldfarb, I.; Gol'dshtein, V.
2009-05-01
The paper concerns intrinsic low-dimensional manifold (ILDM) method suggested in [Maas U, Pope SB. Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, combustion and flame 1992;88:239-64] for dimension reduction of models describing kinetic processes. It has been shown in a number of publications [Goldfarb I, Gol'dshtein V, Maas U. Comparative analysis of two asymptotic approaches based on integral manifolds. IMA J Appl Math 2004;69:353-74; Kaper HG, Kaper TJ, Asymptotic analysis of two reduction methods for systems of chemical reactions. Phys D 2002;165(1-2):66-93; Rhodes C, Morari M, Wiggins S. Identification of the low order manifolds: validating the algorithm of Maas and Pope. Chaos 1999;9(1):108-23] that the ILDM-method works successfully and the intrinsic low-dimensional manifolds belong to a small vicinity of invariant slow manifolds. The ILDM-method has a number of disadvantages. One of them is appearance of so-called "ghost"-manifolds, which do not have connection to the system dynamics [Borok S, Goldfarb I, Gol'dshtein V. "Ghost" ILDM - manifolds and their discrimination. In: Twentieth Annual Symposium of the Israel Section of the Combustion Institute, Beer-Sheva, Israel; 2004. p. 55-7; Borok S, Goldfarb I, Gol'dshtein V. About non-coincidence of invariant manifolds and intrinsic low-dimensional manifolds (ILDM). CNSNS 2008;71:1029-38; Borok S, Goldfarb I, Gol'dshtein V, Maas U. In: Gorban AN, Kazantzis N, Kevrekidis YG, Ottinger HC, Theodoropoulos C, editors. "Ghost" ILDM-manifolds and their identification: model reduction and coarse-graining approaches for multiscale phenomena. Berlin-Heidelberg-New York: Springer; 2006. p. 55-80; Borok S, Goldfarb I, Gol'dshtein V. On a modified version of ILDM method and its asymptotic analysis. IJPAM 2008; 44(1): 125-50; Bykov V, Goldfarb I, Gol'dshtein V, Maas U. On a modified version of ILDM approach: asymptotic analysis based on integral manifolds. IMA J Appl Math 2006
Black-Box Optimization Using Geodesics in Statistical Manifolds
Directory of Open Access Journals (Sweden)
Jérémy Bensadon
2015-01-01
Full Text Available Information geometric optimization (IGO is a general framework for stochastic optimization problems aiming at limiting the influence of arbitrary parametrization choices: the initial problem is transformed into the optimization of a smooth function on a Riemannian manifold, defining a parametrization-invariant first order differential equation and, thus, yielding an approximately parametrization-invariant algorithm (up to second order in the step size. We define the geodesic IGO update, a fully parametrization-invariant algorithm using the Riemannian structure, and we compute it for the manifold of Gaussians, thanks to Noether’s theorem. However, in similar algorithms, such as CMA-ES (Covariance Matrix Adaptation - Evolution Strategy and xNES (exponential Natural Evolution Strategy, the time steps for the mean and the covariance are decoupled. We suggest two ways of doing so: twisted geodesic IGO (GIGO and blockwise GIGO. Finally, we show that while the xNES algorithm is not GIGO, it is an instance of blockwise GIGO applied to the mean and covariance matrix separately. Therefore, xNES has an almost parametrization-invariant description.
Hildebrand, Richard J.; Wozniak, John J.
2001-01-01
A compressed gas storage cell interconnecting manifold including a thermally activated pressure relief device, a manual safety shut-off valve, and a port for connecting the compressed gas storage cells to a motor vehicle power source and to a refueling adapter. The manifold is mechanically and pneumatically connected to a compressed gas storage cell by a bolt including a gas passage therein.
Existence and regularity of minimal surfaces on Riemannian manifolds (MN-27)
Pitts, Jon T
2014-01-01
Mathematical No/ex, 27 Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
A Dynamic Laplacian for Identifying Lagrangian Coherent Structures on Weighted Riemannian Manifolds
Froyland, Gary; Kwok, Eric
2017-06-01
Transport and mixing in dynamical systems are important properties for many physical, chemical, biological, and engineering processes. The detection of transport barriers for dynamics with general time dependence is a difficult, but important problem, because such barriers control how rapidly different parts of phase space (which might correspond to different chemical or biological agents) interact. The key factor is the growth of interfaces that partition phase space into separate regions. The paper Froyland (Nonlinearity 28(10):3587-3622, 2015) introduced the notion of dynamic isoperimetry: the study of sets with persistently small boundary size (the interface) relative to enclosed volume, when evolved by the dynamics. Sets with this minimal boundary size to volume ratio were identified as level sets of dominant eigenfunctions of a dynamic Laplace operator. In this present work we extend the results of Froyland (Nonlinearity 28(10):3587-3622, 2015) to the situation where the dynamics (1) is not necessarily volume preserving, (2) acts on initial agent concentrations different from uniform concentrations, and (3) occurs on a possibly curved phase space. Our main results include generalised versions of the dynamic isoperimetric problem, the dynamic Laplacian, Cheeger's inequality, and the Federer-Fleming theorem. We illustrate the computational approach with some simple numerical examples.
Directory of Open Access Journals (Sweden)
Ruslan A. Sharipov
2002-01-01
keeps orthogonality to the trajectories of all its points. Geodesic lines correspond to the motion of free particles if the points of hypersurface are treated as physical entities obeying Newton's second law. An attempt to introduce some external force F acting on the points of moving hypersurface in Bonnet construction leads to the theory of dynamical systems admitting a normal shift. As appears in this theory, the force field F of dynamical system should satisfy some system of partial differential equations. Recently, this system of equations was integrated, and explicit formula for F was obtained. But this formula is local. The main goal of this paper is to reveal global geometric structures associated with local expressions for F given by explicit formula.
Candela, Anna Maria; Sánchez, Miguel
2013-01-01
Recently, classical results on completeness of trajectories of Hamiltonian systems obtained at the beginning of the seventies, have been revisited, improved and applied to Lorentzian Geometry. Our aim here is threefold: to give explicit proofs of some technicalities in the background of the specialists, to show that the introduced tools allow to obtain more results for the completeness of the trajectories, and to apply these results to the completeness of spacetimes that generalize classical plane and pp-waves.
Zhang, Zhenyue; Wang, Jing; Zha, Hongyuan
2012-02-01
Manifold learning algorithms seek to find a low-dimensional parameterization of high-dimensional data. They heavily rely on the notion of what can be considered as local, how accurately the manifold can be approximated locally, and, last but not least, how the local structures can be patched together to produce the global parameterization. In this paper, we develop algorithms that address two key issues in manifold learning: 1) the adaptive selection of the local neighborhood sizes when imposing a connectivity structure on the given set of high-dimensional data points and 2) the adaptive bias reduction in the local low-dimensional embedding by accounting for the variations in the curvature of the manifold as well as its interplay with the sampling density of the data set. We demonstrate the effectiveness of our methods for improving the performance of manifold learning algorithms using both synthetic and real-world data sets.
Almost-Riemannian Geometry on Lie Groups
Ayala, Victor; Jouan, Philippe
2015-01-01
A simple Almost-Riemmanian Structure on a Lie group G is defined by a linear vector field and dim(G)-1 left-invariant ones. We state results about the singular locus, the abnormal extremals and the desingularization of such ARS's, and these results are illustrated by examples on the 2D affine and the Heisenberg groups.These ARS's are extended in two ways to homogeneous spaces, and a necessary and sufficient condition for an ARS on a manifold to be equivalent to a general ARS on a homogeneous ...
Ensemble manifold regularization.
Geng, Bo; Tao, Dacheng; Xu, Chao; Yang, Linjun; Hua, Xian-Sheng
2012-06-01
We propose an automatic approximation of the intrinsic manifold for general semi-supervised learning (SSL) problems. Unfortunately, it is not trivial to define an optimization function to obtain optimal hyperparameters. Usually, cross validation is applied, but it does not necessarily scale up. Other problems derive from the suboptimality incurred by discrete grid search and the overfitting. Therefore, we develop an ensemble manifold regularization (EMR) framework to approximate the intrinsic manifold by combining several initial guesses. Algorithmically, we designed EMR carefully so it 1) learns both the composite manifold and the semi-supervised learner jointly, 2) is fully automatic for learning the intrinsic manifold hyperparameters implicitly, 3) is conditionally optimal for intrinsic manifold approximation under a mild and reasonable assumption, and 4) is scalable for a large number of candidate manifold hyperparameters, from both time and space perspectives. Furthermore, we prove the convergence property of EMR to the deterministic matrix at rate root-n. Extensive experiments over both synthetic and real data sets demonstrate the effectiveness of the proposed framework.
Model Transport: Towards Scalable Transfer Learning on Manifolds
DEFF Research Database (Denmark)
Freifeld, Oren; Hauberg, Søren; Black, Michael J.
2014-01-01
“commutes” with learning. Consequently, our compact framework, applicable to a large class of manifolds, is not restricted by the size of either the training or test sets. We demonstrate the approach by transferring PCA and logistic-regression models of real-world data involving 3D shapes and image......We consider the intersection of two research fields: transfer learning and statistics on manifolds. In particular, we consider, for manifold-valued data, transfer learning of tangent-space models such as Gaussians distributions, PCA, regression, or classifiers. Though one would hope to simply use...... ordinary Rn-transfer learning ideas, the manifold structure prevents it. We overcome this by basing our method on inner-product-preserving parallel transport, a well-known tool widely used in other problems of statistics on manifolds in computer vision. At first, this straightforward idea seems to suffer...
A plug with infinite order and some exotic 4-manifolds
Tange, Motoo
2012-01-01
Every exotic pair in 4-dimension is obtained each other by twisting a {\\it cork} or {\\it plug} which are codimension 0 submanifolds embedded in the 4-manifolds. The twist was an involution on the boundary of the submanifold. We define cork (or plug) with order $p\\in {\\Bbb N}\\cup \\{\\infty\\}$ and show there exists a plug with infinite order. Furthermore we show twisting $(P,\\varphi^2)$ gives to enlargements of $P$ compact exotic manifolds with boundary.
Webs of Lagrangian Tori in Projective Symplectic Manifolds
Hwang, Jun-Muk
2012-01-01
For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperk\\"ahler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville's. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt's theory of subnormal subgroups.
Quasi-rigidity of hyperbolic 3-manifolds and scattering theory
Borthwick, D; Taylor, E; Borthwick, David; Rae, Alan Mc; Taylor, Edward
1996-01-01
Take two isomorphic convex co-compact co-infinite volume Kleinian groups, whose regular sets are diffeomorphic. The quotient of hyperbolic 3-space by these groups gives two hyperbolic 3-manifolds whose scattering operators may be compared. We prove that the operator norm of the difference between the scattering operators is small, then the groups are related by a coorespondingly small quasi-conformal deformation. This in turn implies that the two hyperbolic 3-manifolds are quasi-isometric.
A Note on Heegaard Splittings of Amalgamated 3-Manifolds
Institute of Scientific and Technical Information of China (English)
Kun DU; Xutao GAO
2011-01-01
Let M be a compact orientable irreducible 3-manifold, and F be an essential connected closed surface in M which cuts M into two manifolds M1 and M2. If Mi has a minimal Heegaard splitting Mi = Vi ∪Hi Wi with d(H1) + d(H2) ≥ 2(g(M1) + g(M2) -g(F)) + 1, then g(M) = g(M1) + g(M2) - g(F).
Adaptive Error Detection Method for P300-based Spelling Using Riemannian Geometry
Directory of Open Access Journals (Sweden)
Attaullah Sahito
2016-11-01
Full Text Available Brain-Computer Interface (BCI systems have be-come one of the valuable research area of ML (Machine Learning and AI based techniques have brought significant change in traditional diagnostic systems of medical diagnosis. Specially; Electroencephalogram (EEG, which is measured electrical ac-tivity of the brain and ionic current in neurons is result of these activities. A brain-computer interface (BCI system uses these EEG signals to facilitate humans in different ways. P300 signal is one of the most important and vastly studied EEG phenomenon that has been studied in Brain Computer Interface domain. For instance, P300 signal can be used in BCI to translate the subject’s intention from mere thoughts using brain waves into actual commands, which can eventually be used to control different electro mechanical devices and artificial human body parts. Since low Signal-to-Noise-Ratio (SNR in P300 is one of the major challenge because concurrently ongoing heterogeneous activities and artifacts of brain creates lots of challenges for doctors to understand the human intentions. In order to address above stated challenge this research proposes a system so called Adaptive Error Detection method for P300-Based Spelling using Riemannian Geometry, the system comprises of three main steps, in first step raw signal is cleaned by preprocessing. In second step most relevant features are extracted using xDAWN spatial filtering along with covariance matrices for handling high dimensional data and in final step elastic net classification algorithm is applied after converting from Riemannian manifold to Euclidean space using tangent space mapping. Results obtained by proposed method are comparable to state-of-the-art methods, as they decrease time drastically; as results suggest six times decrease in time and perform better during the inter-session and inter-subject variability.
Munkres, James R
1997-01-01
A readable introduction to the subject of calculus on arbitrary surfaces or manifolds. Accessible to readers with knowledge of basic calculus and linear algebra. Sections include series of problems to reinforce concepts.
A quantitative version of Steinhaus’ theorem for compact, connected, rank-one symmetric spaces
Oliveira Filho, F.M. de; Vallentin, F.
2010-01-01
Let $d_1$, $d_2$, ... be a sequence of positive numbers that converges to zero. A generalization of Steinhaus' theorem due to Weil implies that, if a subset of a homogeneous Riemannian manifold has no pair of points at distances $d_1$, $d_2$, ... from each other, then it has to have measure zero. We
Hierarchical manifold learning.
Bhatia, Kanwal K; Rao, Anil; Price, Anthony N; Wolz, Robin; Hajnal, Jo; Rueckert, Daniel
2012-01-01
We present a novel method of hierarchical manifold learning which aims to automatically discover regional variations within images. This involves constructing manifolds in a hierarchy of image patches of increasing granularity, while ensuring consistency between hierarchy levels. We demonstrate its utility in two very different settings: (1) to learn the regional correlations in motion within a sequence of time-resolved images of the thoracic cavity; (2) to find discriminative regions of 3D brain images in the classification of neurodegenerative disease,
Joyce, Dominic
2009-01-01
Manifolds without boundary, and manifolds with boundary, are universally known in Differential Geometry, but manifolds with corners (locally modelled on [0,\\infty)^k x R^{n-k}) have received comparatively little attention. The basic definitions in the subject are not agreed upon, there are several inequivalent definitions in use of manifolds with corners, of boundary, and of smooth map, depending on the applications in mind. We present a theory of manifolds with corners which includes a new notion of smooth map f : X --> Y. Compared to other definitions, our theory has the advantage of giving a category Man^c of manifolds with corners which is particularly well behaved as a category: it has products and direct products, boundaries behave in a functorial way, and there are simple conditions for the existence of fibre products X x_Z Y in Man^c. Our theory is tailored to future applications in Symplectic Geometry, and is part of a project to describe the geometric structure on moduli spaces of J-holomorphic curv...
An extension of the (1,2)-symplectic property for f-structures on flag manifolds
Energy Technology Data Exchange (ETDEWEB)
Cohen, N [Instituto de Matematica, Estatistica e Computacao Cientifica, Campinas (Brazil); Pinzon, S [Universidad Industrial de Santander, Bucaramanga (Colombia)
2008-06-30
The (1,1)-symplectic property for f-structures on a complex Riemannian manifold M is a natural extension of the (1,2)-symplectic property for almost-complex structures on M, and arises in the analysis of complex harmonic maps with values in M. A characterization of this property in combinatorial terms is known only for almost-complex structures or when M is the classical flag manifold F(n). In this paper, we remove these restrictions by considering an intersection graph defined in terms of the corresponding root system. We prove that the f-structure is (1,1)-symplectic exactly when the intersection graph is locally transitive. Our intersection graph construction may be helpful in characterizing many other Kaehler-like properties on complex flag manifolds.
Microlocal properties of scattering matrices for Schr\\"odinger equations on scattering manifolds
Ito, Kenichi
2011-01-01
Let $M$ be a scattering manifold, i.e., a Riemannian manifold with asymptotically conic structure, and let $H$ be a Schr\\"odinger operator on $M$. We can construct a natural time-dependent scattering theory for $H$ with a suitable reference system, and the scattering matrix is defined accordingly. We here show the scattering matrices are Fourier integral operators associated to a canonical transform on the boundary manifold generated by the geodesic flow. In particular, we learn that the wave front sets are mapped according to the canonical transform. These results are generalizations of a theorem by Melrose and Zworski, but the framework and the proof are quite different. These results may be considered as generalizations or refinements of the classical off-diagonal smoothness of the scattering matrix for 2-body quantum scattering on Euclidean spaces.
Liu, Yang; Liu, Yan; Chan, Keith C C; Hua, Kien A
2014-12-01
In this brief, we present a novel supervised manifold learning framework dubbed hybrid manifold embedding (HyME). Unlike most of the existing supervised manifold learning algorithms that give linear explicit mapping functions, the HyME aims to provide a more general nonlinear explicit mapping function by performing a two-layer learning procedure. In the first layer, a new clustering strategy called geodesic clustering is proposed to divide the original data set into several subsets with minimum nonlinearity. In the second layer, a supervised dimensionality reduction scheme called locally conjugate discriminant projection is performed on each subset for maximizing the discriminant information and minimizing the dimension redundancy simultaneously in the reduced low-dimensional space. By integrating these two layers in a unified mapping function, a supervised manifold embedding framework is established to describe both global and local manifold structure as well as to preserve the discriminative ability in the learned subspace. Experiments on various data sets validate the effectiveness of the proposed method.
Tensors and Riemannian geometry with applications to differential equations
Ibragimov, Nail H
2015-01-01
This graduate textbook begins by introducing Tensors and Riemannian Spaces, and then elaborates their application in solving second-order differential equations, and ends with introducing theory of relativity and de Sitter space. Based on 40 years of teaching experience, the author compiles a well-developed collection of examples and exercises to facilitate the reader’s learning.
Manifolds, sheaves, and cohomology
Wedhorn, Torsten
2016-01-01
This book explains techniques that are essential in almost all branches of modern geometry such as algebraic geometry, complex geometry, or non-archimedian geometry. It uses the most accessible case, real and complex manifolds, as a model. The author especially emphasizes the difference between local and global questions. Cohomology theory of sheaves is introduced and its usage is illustrated by many examples. Content Topological Preliminaries - Algebraic Topological Preliminaries - Sheaves - Manifolds - Local Theory of Manifolds - Lie Groups - Torsors and Non-abelian Cech Cohomology - Bundles - Soft Sheaves - Cohomology of Complexes of Sheaves - Cohomology of Sheaves of Locally Constant Functions - Appendix: Basic Topology, The Language of Categories, Basic Algebra, Homological Algebra, Local Analysis Readership Graduate Students in Mathematics / Master of Science in Mathematics About the Author Prof. Dr. Torsten Wedhorn, Department of Mathematics, Technische Universität Darmstadt, Germany.
Manifold Learning by Graduated Optimization.
Gashler, M; Ventura, D; Martinez, T
2011-12-01
We present an algorithm for manifold learning called manifold sculpting , which utilizes graduated optimization to seek an accurate manifold embedding. An empirical analysis across a wide range of manifold problems indicates that manifold sculpting yields more accurate results than a number of existing algorithms, including Isomap, locally linear embedding (LLE), Hessian LLE (HLLE), and landmark maximum variance unfolding (L-MVU), and is significantly more efficient than HLLE and L-MVU. Manifold sculpting also has the ability to benefit from prior knowledge about expected results.
Manifold Regularized Reinforcement Learning.
Li, Hongliang; Liu, Derong; Wang, Ding
2017-01-27
This paper introduces a novel manifold regularized reinforcement learning scheme for continuous Markov decision processes. Smooth feature representations for value function approximation can be automatically learned using the unsupervised manifold regularization method. The learned features are data-driven, and can be adapted to the geometry of the state space. Furthermore, the scheme provides a direct basis representation extension for novel samples during policy learning and control. The performance of the proposed scheme is evaluated on two benchmark control tasks, i.e., the inverted pendulum and the energy storage problem. Simulation results illustrate the concepts of the proposed scheme and show that it can obtain excellent performance.
Target-local Gromov compactness
Fish, Joel W
2009-01-01
We prove a version of Gromov's compactness theorem for pseudo-holomorphic curves which holds locally in the target symplectic manifold. This result applies to sequences of curves with an unbounded number of free boundary components, and in families of degenerating target manifolds which have unbounded geometry (e.g. no uniform energy threshold). Core elements of the proof regard curves as submanifolds (rather than maps) and then adapt methods from the theory of minimal surfaces.
The Einstein-Λ flow on product manifolds
Fajman, David; Kröncke, Klaus
2016-12-01
We consider the vacuum Einstein flow with a positive cosmological constant {{Λ }} on spatial manifolds of product form M={M}1× {M}2. In dimensions n=\\dim M≥slant 4 we show the existence of continuous families of recollapsing models whenever at least one of the factors M 1 or M 2 admits a Riemannian Einstein metric with positive Einstein constant. We moreover show that these families belong to larger continuous families with models that have two complete time directions, i.e. do not recollapse. Complementarily, we show that whenever no factor has positive curvature, then any model in the product class expands in one time direction and collapses in the other. In particular, positive curvature of one factor is a necessary criterion for recollapse within this class. Finally, we relate our results to the instability of the Nariai solution in three spatial dimensions and point out why a similar construction of recollapsing models in that dimension fails. The present results imply that there exist different classes of initial data which exhibit fundamentally different types of long-time behavior under the Einstein-{{Λ }} flow whenever the spatial dimension is strictly larger than three. Moreover, this behavior is related to the spatial topology through the existence of Riemannian Einstein metrics of positive curvature.
Some connectedness problems in positively curved Finsler manifolds
Peter, Ioan Radu
2009-01-01
This paper studies some connectedness problems under the positivity hypothesis of various curvatures ( k-Ricci and flag curvature). Our approach uses Morse Theory for general end conditions (see [Ioan Radu Peter, The Morse index theorem where the ends are submanifolds in Finsler geometry, Houston J. Math. 32 (4) (2006) 995-1009]). Some previous results related to the flag curvature were obtained in [Ioan Radu Peter, A connectedness principle in positively curved Finsler manifolds, in: H. Shimada, S. Sabau (Eds.), Advanced Studies in Pure Mathematics, Finsler Geometry, Sapporo 2005-In Memory of Makoto Matsumoto, Mathematical Society of Japan, 2007]. Some results from Riemannian geometry are extended to the Finsler category also. The Finsler setting is much more complicated and the difference between Finsler and Riemann settings will be emphasized during the paper.
The Einstein flow with positive cosmological constant on product manifolds
Fajman, David
2016-01-01
We consider the vacuum Einstein flow with a positive cosmological constant on spatial manifolds of product form. In spatial dimension at least four we show the existence of continuous families of recollapsing models whenever at least one of the factors or admits a Riemannian Einstein metric with positive Einstein constant. We moreover show that these families belong to larger continuous families with models that have two complete time directions, i.e. do not recollapse. Complementarily, we show that whenever no factor has positive curvature, then any model in the product class expands in one time direction and collapses in the other. In particular, positive curvature of one factor is a necessary criterion for recollapse within this class. Finally, we relate our results to the instability of the Nariai solution in three spatial dimensions and point out why a similar construction of recollapsing models in that dimension fails. The present results imply that there exist different classes of initial data which ex...
Black Strings, Black Rings and State-space Manifold
Bellucci, Stefano
2011-01-01
State-space geometry is considered, for diverse three and four parameter non-spherical horizon rotating black brane configurations, in string theory and $M$-theory. We have explicitly examined the case of unit Kaluza-Klein momentum $D_1D_5P$ black strings, circular strings, small black rings and black supertubes. An investigation of the state-space pair correlation functions shows that there exist two classes of brane statistical configurations, {\\it viz.}, the first category divulges a degenerate intrinsic equilibrium basis, while the second yields a non-degenerate, curved, intrinsic Riemannian geometry. Specifically, the solutions with finitely many branes expose that the two charged rotating $D_1D_5$ black strings and three charged rotating small black rings consort real degenerate state-space manifolds. Interestingly, arbitrary valued $M_5$-dipole charged rotating circular strings and Maldacena Strominger Witten black rings exhibit non-degenerate, positively curved, comprehensively regular state-space con...
Congedo, Marco; Barachant, Alexandre
2015-01-01
Currently the Riemannian geometry of symmetric positive definite (SPD) matrices is gaining momentum as a powerful tool in a wide range of engineering applications such as image, radar and biomedical data signal processing. If the data is not natively represented in the form of SPD matrices, typically we may summarize them in such form by estimating covariance matrices of the data. However once we manipulate such covariance matrices on the Riemannian manifold we lose the representation in the original data space. For instance, we can evaluate the geometric mean of a set of covariance matrices, but not the geometric mean of the data generating the covariance matrices, the space of interest in which the geometric mean can be interpreted. As a consequence, Riemannian information geometry is often perceived by non-experts as a "black-box" tool and this perception prevents a wider adoption in the scientific community. Hereby we show that we can overcome this limitation by constructing a special form of SPD matrix embedding both the covariance structure of the data and the data itself. Incidentally, whenever the original data can be represented in the form of a generic data matrix (not even square), this special SPD matrix enables an exhaustive and unique description of the data up to second-order statistics. This is achieved embedding the covariance structure of both the rows and columns of the data matrix, allowing naturally a wide range of possible applications and bringing us over and above just an interpretability issue. We demonstrate the method by manipulating satellite images (pansharpening) and event-related potentials (ERPs) of an electroencephalography brain-computer interface (BCI) study. The first example illustrates the effect of moving along geodesics in the original data space and the second provides a novel estimation of ERP average (geometric mean), showing that, in contrast to the usual arithmetic mean, this estimation is robust to outliers. In
Yamabe flow on Berwald manifolds
Azami, Shahroud; Razavi, Asadollah
2015-12-01
Studying the geometric flow plays a powerful role in mathematics and physics. We introduce the Yamabe flow on Finsler manifolds and we will prove the existence and uniqueness for solution of Yamabe flow on Berwald manifolds.
Gómez, Gerard; Barrabés Vera, Esther
2011-01-01
The term Space Manifold Dynamics (SMD) has been proposed for encompassing the various applications of Dynamical Systems methods to spacecraft mission analysis and design, ranging from the exploitation of libration orbits around the collinear Lagrangian points to the design of optimal station-keeping and eclipse avoidance manoeuvres or the determination of low energy lunar and interplanetary transfers
Thinking Outside the Euclidean Box: Riemannian Geometry and Inter-Temporal Decision-Making.
Mishra, Himanshu; Mishra, Arul
2016-01-01
Inter-temporal decisions involves assigning values to various payoffs occurring at different temporal distances. Past research has used different approaches to study these decisions made by humans and animals. For instance, considering that people discount future payoffs at a constant rate (e.g., exponential discounting) or at variable rate (e.g., hyperbolic discounting). In this research, we question the widely assumed, but seldom questioned, notion across many of the existing approaches that the decision space, where the decision-maker perceives time and monetary payoffs, is a Euclidean space. By relaxing the rigid assumption of Euclidean space, we propose that the decision space is a more flexible Riemannian space of Constant Negative Curvature. We test our proposal by deriving a discount function, which uses the distance in the Negative Curvature space instead of Euclidean temporal distance. The distance function includes both perceived values of time as well as money, unlike past work which has considered just time. By doing so we are able to explain many of the empirical findings in inter-temporal decision-making literature. We provide converging evidence for our proposal by estimating the curvature of the decision space utilizing manifold learning algorithm and showing that the characteristics (i.e., metric properties) of the decision space resembles those of the Negative Curvature space rather than the Euclidean space. We conclude by presenting new theoretical predictions derived from our proposal and implications for how non-normative behavior is defined.
Thinking Outside the Euclidean Box: Riemannian Geometry and Inter-Temporal Decision-Making.
Directory of Open Access Journals (Sweden)
Himanshu Mishra
Full Text Available Inter-temporal decisions involves assigning values to various payoffs occurring at different temporal distances. Past research has used different approaches to study these decisions made by humans and animals. For instance, considering that people discount future payoffs at a constant rate (e.g., exponential discounting or at variable rate (e.g., hyperbolic discounting. In this research, we question the widely assumed, but seldom questioned, notion across many of the existing approaches that the decision space, where the decision-maker perceives time and monetary payoffs, is a Euclidean space. By relaxing the rigid assumption of Euclidean space, we propose that the decision space is a more flexible Riemannian space of Constant Negative Curvature. We test our proposal by deriving a discount function, which uses the distance in the Negative Curvature space instead of Euclidean temporal distance. The distance function includes both perceived values of time as well as money, unlike past work which has considered just time. By doing so we are able to explain many of the empirical findings in inter-temporal decision-making literature. We provide converging evidence for our proposal by estimating the curvature of the decision space utilizing manifold learning algorithm and showing that the characteristics (i.e., metric properties of the decision space resembles those of the Negative Curvature space rather than the Euclidean space. We conclude by presenting new theoretical predictions derived from our proposal and implications for how non-normative behavior is defined.
Manifold Insulation for Solar Collectors
1982-01-01
Results of computer analysis of effects of various manifold insulation detailed in 23-page report show that if fluid is distributed to and gathered from array of solar collectors by external rather than internal manifold, effectiveness of manifold insulation has major influence on efficiency. Report describes required input data and presents equations that govern computer model. Provides graphs comparing collector efficiencies for representative manifold sizes and insulations.
Pulse Distributing Manifold; Pulse Distributing Manifold
Energy Technology Data Exchange (ETDEWEB)
Schutting, Eberhard [Technische Univ. Graz (Austria); Sams, Theodor [AVL List GmbH, Graz (Austria); Glensvig, Michael [Forschungsgesellschaft mbH, Graz (AT). Kompetenzzentrum ' ' Das virtuelle Fahrzeug' ' (VIF)
2011-07-01
The Pulse Distributing Manifold is a new charge exchange method for turbocharged diesel engines with exhaust gas recirculation (EGR). The method is characterized in that the EGR mass flow is not diverted from the exhaust gas mass flow continuously, but over time broken into sub-streams. The temporal interruption is achieved by two phase-shifted outlet valves which are connected via separate manifolds only with the turbocharger or only with the EGR path. The time points of valve opening are chosen such that the turbocharger and the aftertreatment process of exhaust gas is perfused by high-energy exhaust gas of the blowdown phase while cooler and less energy-rich exhaust gas of the exhaust period is used for the exhaust gas recirculation. This increases the enthalpy for the turbocharger and the temperature for the exhaust gas treatment, while the cooling efficiency at the EGR cooler is reduced. The elimination of the continuous EGR valve has a positive effect on pumping losses. The principle functioning and the potential of this system could be demonstrated by means of a concept study using one-dimensional simulations. Without disadvantages in fuel consumption for the considered commercial vehicle engine, a reduction the EGR cooler performance by 15 % and an increase in exhaust temperature of 35 K could be achieved. The presented charge exchange method was developed, evaluated and patented within the scope of the research program 'K2-mobility' of the project partners AVL (Mainz, Federal Republic of Germany) and University of Technology Graz (Austria). The research project 'K2-Mobility' is supported by the competence center 'The virtual vehicle' Forschungsgesellschaft mbH (Graz, Austria).
Supersymmetric Proof of the Hirzebruch-Riemann-Roch Theorem for Non-Kähler Manifolds
Directory of Open Access Journals (Sweden)
Andrei V. Smilga
2012-01-01
Full Text Available We present the proof of the HRR theorem for a generic complex compact manifold by evaluating the functional integral for the Witten index of the appropriate supersymmetric quantum mechanical system.
Supervised learning for neural manifold using spatiotemporal brain activity
Kuo, Po-Chih; Chen, Yong-Sheng; Chen, Li-Fen
2015-12-01
Objective. Determining the means by which perceived stimuli are compactly represented in the human brain is a difficult task. This study aimed to develop techniques for the construction of the neural manifold as a representation of visual stimuli. Approach. We propose a supervised locally linear embedding method to construct the embedded manifold from brain activity, taking into account similarities between corresponding stimuli. In our experiments, photographic portraits were used as visual stimuli and brain activity was calculated from magnetoencephalographic data using a source localization method. Main results. The results of 10 × 10-fold cross-validation revealed a strong correlation between manifolds of brain activity and the orientation of faces in the presented images, suggesting that high-level information related to image content can be revealed in the brain responses represented in the manifold. Significance. Our experiments demonstrate that the proposed method is applicable to investigation into the inherent patterns of brain activity.
Quadrature rules and distribution of points on manifolds
Brandolini, Luca; Colzani, Leonardo; Gigante, Giacomo; Seri, Raffaello; Travaglini, Giancarlo
2010-01-01
We study the error in quadrature rules on a compact manifold. As in the Koksma-Hlawka inequality, we consider a discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.
Beckner Inequality on Finite- and Infinite-Dimensional Manifolds
Institute of Scientific and Technical Information of China (English)
Pingji DENG; Fengyu WANG
2006-01-01
By using the dimension-free Harnack inequality, the coupling method, and Bakry-Emery's argument, some explicit lower bounds are presented for the constant of the Beckner type inequality on compact manifolds. As applications, the Beckner inequality and the transportation cost inequality are established for a class of continuous spin systems.In particular, some results in [1, 2] are generalized.
Eigenvalues of the Dirac operator on manifolds with boundary
Energy Technology Data Exchange (ETDEWEB)
Hijazi, O. [Inst. Elie Cartan, Univ. Henri Poincare, Nancy (France); Montiel, S. [Dept. de Geometria y Topologia, Universidad de Granada (Spain); Zhang, X. [Inst. of Mathematics, Academy of Mathematics and Systems Sciences, Beijing (China)
2001-07-01
Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. For the local boundary conditions, limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary. (orig.)
Canonical metrics on complex manifold
Institute of Scientific and Technical Information of China (English)
YAU Shing-Tung
2008-01-01
@@ Complex manifolds are topological spaces that are covered by coordinate charts where the Coordinate changes are given by holomorphic transformations. For example, Riemann surfaces are one dimensional complex manifolds. In order to understand complex manifolds, it is useful to introduce metrics that are compatible with the complex structure. In general, we should have a pair (M, ds2M) where ds2M is the metric. The metric is said to be canonical if any biholomorphisms of the complex manifolds are automatically isometries. Such metrics can naturally be used to describe invariants of the complex structures of the manifold.
Canonical metrics on complex manifold
Institute of Scientific and Technical Information of China (English)
YAU; Shing-Tung(Yau; S.-T.)
2008-01-01
Complex manifolds are topological spaces that are covered by coordinate charts where the coordinate changes are given by holomorphic transformations.For example,Riemann surfaces are one dimensional complex manifolds.In order to understand complex manifolds,it is useful to introduce metrics that are compatible with the complex structure.In general,we should have a pair(M,ds~2_M)where ds~2_M is the metric.The metric is said to be canonical if any biholomorphisms of the complex manifolds are automatically isometries.Such metrics can naturally be used to describe invariants of the complex structures of the manifold.
Invariant manifolds and global bifurcations.
Guckenheimer, John; Krauskopf, Bernd; Osinga, Hinke M; Sandstede, Björn
2015-09-01
Invariant manifolds are key objects in describing how trajectories partition the phase spaces of a dynamical system. Examples include stable, unstable, and center manifolds of equilibria and periodic orbits, quasiperiodic invariant tori, and slow manifolds of systems with multiple timescales. Changes in these objects and their intersections with variation of system parameters give rise to global bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families of dynamical systems also play a prominent role in dynamical systems theory. Much progress has been made in developing theory and computational methods for invariant manifolds during the past 25 years. This article highlights some of these achievements and remaining open problems.
Riemannian geometrical constraints on magnetic vortex filaments in plasmas
de Andrade, L. C. Garcia
2005-01-01
Two theorems on the Riemannian geometrical constraints on vortex magnetic filaments acting as dynamos in (MHD) flows are presented. The use of Gauss-Mainard-Codazzi equations allows us to investigate in detail the influence of curvature and torsion of vortex filaments in the MHD dynamos. This application follows closely previous applications to Heisenberg spin equation to the investigations in magnetohydrostatics given by Schief (Plasma Physics J. 10, 7, 2677 (2003)). The Lorentz force on vor...
Non-negative Ricci curvature on closed manifolds under Ricci flow
Maximo, Davi
2009-01-01
In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \\cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result B\\"ohm and Wilking have for dimensions twelve and above, \\cite{BW}. Moreover, the manifolds constructed here are \\Kahler manifolds and relate to a question raised by Xiuxiong Chen in \\cite{XC}, \\cite{XCL}.
Iwasawa nilpotency degree of non compact symmetric cosets in N-extended supergravity
Energy Technology Data Exchange (ETDEWEB)
Cacciatori, S.L. [Dipartimento di Scienze ed Alta Tecnologia, Universita degli Studi dell' Insubria, Como (Italy); INFN, Sezione di Milano (Italy); Cerchiai, B.L. [INFN, Sezione di Milano (Italy); Dipartimento di Matematica, Universita degli Studi di Milano (Italy); Ferrara, S. [Physics Department, Theory Unit, CERN, Geneva (Switzerland); INFN - Laboratori Nazionali di Frascati (Italy); Department of Physics and Astronomy, University of California, Los Angeles, CA (United States); Marrani, A. [Instituut voor Theoretische Fysica, KU Leuven (Belgium)
2014-04-01
We analyze the polynomial part of the Iwasawa realization of the coset representative of non compact symmetric Riemannian spaces. We start by studying the role of Kostant's principal SU(2){sub P} subalgebra of simple Lie algebras, and how it determines the structure of the nilpotent subalgebras. This allows us to compute the maximal degree of the polynomials for all faithful representations of Lie algebras. In particular the metric coefficients are related to the scalar kinetic terms while the representation of electric and magnetic charges is related to the coupling of scalars to vector field strengths as they appear in the Lagrangian. We consider symmetric scalar manifolds in N-extended supergravity in various space-time dimensions, elucidating various relations with the underlying Jordan algebras and normed Hurwitz algebras. For magic supergravity theories, our results are consistent with the Tits-Satake projection of symmetric spaces and the nilpotency degree turns out to depend only on the space-time dimension of the theory. These results should be helpful within a deeper investigation of the corresponding supergravity theory, e.g. in studying ultraviolet properties of maximal supergravity in various dimensions. (Copyright copyright 2014 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
On the trace-manifold generated by the deformations of a body-manifold
Directory of Open Access Journals (Sweden)
Boja Nicolae
2003-01-01
Full Text Available In this paper, concerned to the study of continuous deformations of material media using some tools of modem differential geometry, a moving frame of Frenet type along the orbits of an one-parameter group acting on a so-called "trace-manifold", M, associated to the deformations, is constructed. The manifold M is defined as an infinite union of non-disjoint compact manifolds, generated by the consecutive positions in the Euclidean affine 3-space of a body-manifold under deformations in a closed time interval. We put in evidence a skew-symmetric band tensor of second order, ω, which describes the deformation in a small neighborhood of any point along the orbits. The non-null components ωi,i+i, (i =1,2, of ω are assimilated as like curvatures at each point of an orbit in the planes generated by the pairs of vectors (ĕi,ĕi+i of a moving frame in M associated to the orbit in a similar way as the Frenet's frame is. Also a formula for the energy of the orbits is given and its relationship with some stiffness matrices is established.
Learning Smooth Pattern Transformation Manifolds
Vural, Elif
2011-01-01
Manifold models provide low-dimensional representations that are useful for processing and analyzing data in a transformation-invariant way. In this paper, we study the problem of learning smooth pattern transformation manifolds from image sets that represent observations of geometrically transformed signals. In order to construct a manifold, we build a representative pattern whose transformations accurately fit various input images. We examine two objectives of the manifold building problem, namely, approximation and classification. For the approximation problem, we propose a greedy method that constructs a representative pattern by selecting analytic atoms from a continuous dictionary manifold. We present a DC (Difference-of-Convex) optimization scheme that is applicable to a wide range of transformation and dictionary models, and demonstrate its application to transformation manifolds generated by rotation, translation and anisotropic scaling of a reference pattern. Then, we generalize this approach to a s...
Holonomy groups of Lorentzian manifolds
Galaev, Anton S
2016-01-01
In this paper, a survey of the recent results about the classification of the connected holonomy groups of the Lorentzian manifolds is given. A simplification of the construction of the Lorentzian metrics with all possible connected holonomy groups is obtained. As the applications, the Einstein equation, Lorentzian manifolds with parallel and recurrent spinor fields, conformally flat Walker metrics and the classification of 2-symmetric Lorentzian manifolds are considered.
Quantum manifolds with classical limit
Hohmann, Manuel; Wohlfarth, Mattias N R
2008-01-01
We propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and the manifold structure of spacetime. In this picture we demonstrate that classical spacetime emerges as a finite-dimensional manifold through the topological identification of all quantum points with identical position expectation value. We speculate on the possible relevance of this geometry to quantum field theory and gravity.
Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds
Bachman, David
2012-01-01
Let M_1 and M_2 be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism $\\phi:\\bdy M_1 \\to \\bdy M_2$. We analyze the relationship between the sets of low genus Heegaard splittings of M_1, M_2, and M, assuming the map \\phi is "sufficiently complicated." This analysis yields counter-examples to the Stabilization Conjecture, a resolution of the higher genus analogue of a conjecture of Gordon, and a result about the uniqueness of expressions of Heegaard splittings as amalgamations.
Analysis, manifolds and physics
Choquet-Bruhat, Y
2000-01-01
Twelve problems have been added to the first edition; four of them are supplements to problems in the first edition. The others deal with issues that have become important, since the first edition of Volume II, in recent developments of various areas of physics. All the problems have their foundations in volume 1 of the 2-Volume set Analysis, Manifolds and Physics. It would have been prohibitively expensive to insert the new problems at their respective places. They are grouped together at the end of this volume, their logical place is indicated by a number of parenthesis following the title.
Daverman, Robert J
2007-01-01
Decomposition theory studies decompositions, or partitions, of manifolds into simple pieces, usually cell-like sets. Since its inception in 1929, the subject has become an important tool in geometric topology. The main goal of the book is to help students interested in geometric topology to bridge the gap between entry-level graduate courses and research at the frontier as well as to demonstrate interrelations of decomposition theory with other parts of geometric topology. With numerous exercises and problems, many of them quite challenging, the book continues to be strongly recommended to eve
Moment-angle manifolds, intersection of quadrics and higher dimensional contact manifolds
Barreto, Yadira; Verjovsky, Alberto
2013-01-01
We construct new examples of contact manifolds in arbitrarily large dimensions. These manifolds which we call quasi moment-angle manifolds, are closely related to the classical moment-angle manifolds.
Incremental Alignment Manifold Learning
Institute of Scientific and Technical Information of China (English)
Zhi Han; De-Yu Meng; Zong-Sen Xu; Nan-Nan Gu
2011-01-01
A new manifold learning method, called incremental alignment method (IAM), is proposed for nonlinear dimensionality reduction of high dimensional data with intrinsic low dimensionality. The main idea is to incrementally align low-dimensional coordinates of input data patch-by-patch to iteratively generate the representation of the entire dataset. The method consists of two major steps, the incremental step and the alignment step. The incremental step incrementally searches neighborhood patch to be aligned in the next step, and the alignment step iteratively aligns the low-dimensional coordinates of the neighborhood patch searched to generate the embeddings of the entire dataset. Compared with the existing manifold learning methods, the proposed method dominates in several aspects: high efficiency, easy out-of-sample extension, well metric-preserving, and averting of the local minima issue. All these properties are supported by a series of experiments performed on the synthetic and real-life datasets. In addition, the computational complexity of the proposed method is analyzed, and its efficiency is theoretically argued and experimentally demonstrated.
Characterizing the Depolarizing Quantum Channel in Terms of Riemannian Geometry
Cafaro, Carlo
2011-01-01
We explore the conceptual usefulness of Riemannian geometric tools induced by the statistical concept of distinguishability in quantifying the effect of a depolarizing channel on quantum states. Specifically, we compare the geometries of the interior of undeformed and deformed Bloch spheres related to density operators on a two-dimensional Hilbert space. We show that randomization emerges geometrically through a smaller infinitesimal quantum line element on the deformed Bloch sphere while the uniform contraction manifests itself via a deformed set of geodesics where the spacial components of the deformed four-Bloch vector are simply the contracted versions of the undeformed Bloch vector components.
DELAUNAY TRIANGULATION METHOD OF CURVED SURFACES BASED ON RIEMANNIAN METRIC
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
A method for quality mesh generation of parametric curved surfaces is proposed. It is shown that the main difference between the proposed method and previous ones is that our meshing process is done completely in the parametric domains with the guarantee of mesh quality. To obtain this aim, the Delaunay method is extended to anisotropic context of 2D domains, and a Riemannian metric map is introduced to remedy the mapping distortion from object space to parametric domain. Compared with previous algorithms, the approach is much simpler, more robust and speedy. The algorithm is implemented and examples for several geometries are presented to demonstrate the efficiency and validity of the method.
Symmetry and Transitive Properties of Monohedral f-triangulations of the Riemannian Sphere
Institute of Scientific and Technical Information of China (English)
Ana M. BREDA; J. M. SIGARRETA
2009-01-01
Here we give the complete description of the symmetry group and transitive properties of the set of all of monohedral triangulations of the Riemannian sphere by f-tilings. We shall also show that each monohedral f-tiling of the Riemannian sphere can be seen, up to a spherical isometry, as the singular set of a spherical isometric folding.
Examples of Sol-Solitons in the Pseudo-Riemannian case
Onda, Kensuke
2011-01-01
This paper provides a study of sol-solitons in the pseudo-Riemannian case. In the Riemannian case, all nontrivial homogeneous sol-soliton are expanding sol-solitons. In this paper, we obtain steady sol-solitons and shrinking sol-solitons in the Lorentzian setting.
Quantum Riemannian geometry of phase space and nonassociativity
Directory of Open Access Journals (Sweden)
Beggs Edwin J.
2017-04-01
Full Text Available Noncommutative or ‘quantum’ differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket while the data for quantum differential forms is a Poisson-compatible connection. We give an introduction to our recent result whereby further classical data such as classical bundles, metrics etc. all become quantised in a canonical ‘functorial’ way at least to 1st order in deformation theory. The theory imposes compatibility conditions between the classical Riemannian and Poisson structures as well as new physics such as typical nonassociativity of the differential structure at 2nd order. We develop in detail the case of ℂℙn where the commutation relations have the canonical form [wi, w̄j] = iλδij similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in λ.
Semiquantisation Functor and Poisson-Riemannian Geometry, I
Beggs, Edwin J
2014-01-01
We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter $\\lambda$, using a functorial approach. The data for quantisation of the cotangent bundle is known to be a Poisson structure and Poisson preconnection and we now show that this data defines to a functor $Q$ from the monoidal category of classical vector bundles equipped with connections to the monodial category of bimodules equipped with bimodule connections over the quantised algebra. We adapt this functor to quantise the wedge product of the exterior algebra and in the Riemannian case, the metric and the Levi-Civita connection. Full metric compatibility requires vanishing of an obstruction in the classical data, expressed in terms of a generalised Ricci 2-form, without which our quantum Levi-Civita connection is still the best possible. We apply the theory to the Schwarzschild black-hole and to Riemann surfaces as examples, as well as verifying our results on the 2D bicrossproduct mod...
Conjectures on counting associative 3-folds in $G_2$-manifolds
Joyce, Dominic
2016-01-01
There is a strong analogy between compact, torsion-free $G_2$-manifolds $(X,\\varphi,*\\varphi)$ and Calabi-Yau 3-folds $(Y,J,g,\\omega)$. We can also generalize $(X,\\varphi,*\\varphi)$ to 'tamed almost $G_2$-manifolds' $(X,\\varphi,\\psi)$, where we compare $\\varphi$ with $\\omega$ and $\\psi$ with $J$. Associative 3-folds in $X$, a special kind of minimal submanifold, are analogous to $J$-holomorphic curves in $Y$. Several areas of Symplectic Geometry -- Gromov-Witten theory, Quantum Cohomology, Lagrangian Floer cohomology, Fukaya categories -- are built using 'counts' of moduli spaces of $J$-holomorphic curves in $Y$, but give an answer depending only on the symplectic manifold $(Y,\\omega)$, not on the (almost) complex structure $J$. We investigate whether it may be possible to define interesting invariants of tamed almost $G_2$-manifolds $(X,\\varphi,\\psi)$ by 'counting' compact associative 3-folds $N\\subset X$, such that the invariants depend only on $\\varphi$, and are independent of the 4-form $\\psi$ used to def...
The Hantzsche-Wendt manifold in cosmic topology
Aurich, R.; Lustig, S.
2014-08-01
The Hantzsche-Wendt space is one of the 17 multiply connected spaces of the three-dimensional Euclidean space {{{E}}^{3}}. It is a compact and orientable manifold which can serve as a model for a spatial finite universe. Since it possesses much fewer matched back-to-back circle pairs on the cosmic microwave background (CMB) sky than the other compact flat spaces, it can escape the detection by a search for matched circle pairs. The suppression of temperature correlations C(\\vartheta ) on large angular scales on the CMB sky is studied. It is shown that the large-scale correlations are of the same order as for the three-torus topology but express a much larger variability. The Hantzsche-Wendt manifold provides a topological possibility with reduced large-angle correlations that can hide from searches for matched back-to-back circle pairs.
Cohomotopy sets of 4-manifolds
Kirby, Robion; Teichner, Peter
2012-01-01
Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. The former completes a project initiated by Steenrod in the 1940's, and the latter provides geometric arguments for and extensions of recent homotopy theoretic results of Larry Taylor. These two results complete the computation of all the cohomotopy sets of closed oriented 4-manifolds and provide a framework for the study of Morse 2-functions on 4-manifolds, a subject that has garnered considerable recent attention.
The topology of certain 3-Sasakian 7-manifolds
DEFF Research Database (Denmark)
A. Hepworth, Richard
2007-01-01
We calculate the integer cohomology ring and stable tangent bundle of a family of compact, 3-Sasakian 7-manifolds constructed by Boyer, Galicki, Mann, and Rees. Previously only the rational cohomology ring was known. The most important part of the cohomology ring is a torsion group that we describe...... explicitly and whose order we compute. There is a surprising connection with the combinatorics of trees....
Ritt's theorem and the Heins map in hyperbolic complex manifolds
Institute of Scientific and Technical Information of China (English)
Marco; Abate; Filippo; Bracci
2005-01-01
Let X be a Kobayashi hyperbolic complex manifold, and assume that X does not contain compact complex submanifolds of positive dimension (e.g., X Stein). We shall prove the following generalization of Ritt's theorem: every holomorphic self-map f: X →X such that f(X) is relatively compact in X has a unique fixed point τ(f) ∈ X, which is attracting. Furthermore, we shall prove that τ(f) depends holomorphically on f in a suitable sense, generalizing results by Heins, Joseph-Kwack and the second author.
Real group orbits on flag manifolds
Akhiezer, Dmitri
2011-01-01
In this survey, we gather together various results on the action of a real form of a complex semisimple Lie group on its flag manifolds. We start with the finiteness theorem of J.Wolf implying that at least one of the orbits is open. We give a new proof of the converse statement for real forms of inner type, essentially due to F.M.Malyshev. Namely, if a real semisimple Lie group of inner type has an open orbit on an algebraic homogeneous space of the complexified group then the homogeneous space is a flag manifold. To prove this, we recall, partly with proofs, some results of A.L.Onishchik on the factorizations of reductive groups. Finally, we discuss the cycle spaces of open orbits and define the crown of a symmetric space of non-compact type. With some exceptions, the cycle space agrees with the crown. We sketch a complex analytic proof of this result, due to G.Fels, A.Huckleberry and J.Wolf.
Spinor fields classification in arbitrary dimensions and new classes of spinor fields on 7-manifolds
Energy Technology Data Exchange (ETDEWEB)
Bonora, L. [International School for Advanced Studies (SISSA),Via Bonomea 265, 34136 Trieste (Italy); Brito, K.P.S. de [CCNH, Universidade Federal do ABC 09210-580,Santo André, SP (Brazil); Rocha, Roldão da [International School for Advanced Studies (SISSA),Via Bonomea 265, 34136 Trieste (Italy); CMCC, Universidade Federal do ABC 09210-580,Santo André, SP (Brazil)
2015-02-11
A classification of spinor fields according to the associated bilinear covariants is constructed in arbitrary dimensions and metric signatures, generalizing Lounesto’s 4D spinor field classification. In such a generalized classification a basic role is played by the geometric Fierz identities. In 4D Minkowski spacetime the standard bilinear covariants can be either null or non-null — with the exception of the current density which is invariably different from zero for physical reasons — and sweep all types of spinor fields, including Dirac, Weyl, Majorana and more generally flagpoles, flag-dipoles and dipole spinor fields. To obtain an analogous classification in higher dimensions we use the Fierz identities, which constrain the covariant bilinears in the spinor fields and force some of them to vanish. A generalized graded Fierz aggregate is moreover obtained in such a context simply from the completeness relation. We analyze the particular and important case of Riemannian 7-manifolds, where the Majorana spinor fields turn out to have a quite special place. In particular, at variance with spinor fields in 4D Minkowski spacetime that are classified in six disjoint classes, spinors in Riemannian 7-manifolds are shown to be classified, according to the bilinear covariants: (a) in just one class, in the real case of Majorana spinors; (b) in four classes, in the most general case. Much like new classes of spinor fields in 4D Minkowski spacetime have been evincing new possibilities in physics, we think these new classes of spinor fields in seven dimensions are, in particular, potential candidates for new solutions in the compactification of supergravity on a seven-dimensional manifold and its exotic versions.
A New Entropy Formula and Gradient Estimates for the Linear Heat Equation on Static Manifold
Directory of Open Access Journals (Sweden)
Abimbola Abolarinwa
2014-08-01
Full Text Available In this paper we prove a new monotonicity formula for the heat equation via a generalized family of entropy functionals. This family of entropy formulas generalizes both Perelman’s entropy for evolving metric and Ni’s entropy on static manifold. We show that this entropy satisfies a pointwise differential inequality for heat kernel. The consequences of which are various gradient and Harnack estimates for all positive solutions to the heat equation on compact manifold.
Global Regularity for the ∂-b-Equation on CR Manifolds of Arbitrary Codimension
Directory of Open Access Journals (Sweden)
Shaban Khidr
2014-01-01
Full Text Available Let M be a C∞ compact CR manifold of CR-codimension l≥1 and CR-dimension n-l in a complex manifold X of complex dimension n≥3. In this paper, assuming that M satisfies condition Y(s for some s with 1≤s≤n-l-1, we prove an L2-existence theorem and global regularity for the solutions of the tangential Cauchy-Riemann equation for (0,s-forms on M.
Non-Riemannian Cosmic Walls as Boundaries of Spinning Matter
Garcia de Andrade, L C
1998-01-01
An example is given of a plane topological defect solution of linearized Einstein-Cartan (EC) field equation representing a cosmic wall boundary of spinning matter. The source of Cartan torsion is composed of two orthogonal lines of static polarized spins bounded by the cosmic plane wall. The Kopczy\\'{n}ski- Obukhov - Tresguerres (KOT) spin fluid stress-energy current coincides with thin planar matter current in the static case. Our solution is similar to Letelier solution of Einstein equation for multiple cosmic strings. Due to this fact we suggest that the lines of spinning matter could be analogous to multiple cosmic spinning string solution in EC theory of gravity. When torsion is turned off a pure Riemannian cosmic wall is obtained.
Haantjes Manifolds and Integrable Systems
Tempesta, Piergiulio
2014-01-01
A general theory of integrable systems is proposed, based on the theory of Haantjes manifolds. We introduce the notion of symplectic-Haantjes manifold (or $\\omega \\mathcal{H}$ manifold), as the natural setting where the notion of integrability can be formulated. We propose an approach to the separation of variables for classical systems, related to the geometry of Haantjes manifolds. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes structure associated with an integrable systems. They enable the additive separation of variables of the Hamilton-Jacobi equation. We also present an application of our approach to the study of some finite-dimensional integrable models, as the H\\'enon-Heiles systems and a stationary reduction of the KdV hierarchy.
Vector Fields on Product Manifolds
Kurz, Stefan
2011-01-01
This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields. (ii) Horizontal and vertical vector fields are naturally isomorphic to smooth families of vector fields defined on the factors. Vector fields are regarded as derivations of the algebra of smooth functions.
Invariant Manifolds and Collective Coordinates
Papenbrock, T
2001-01-01
We introduce suitable coordinate systems for interacting many-body systems with invariant manifolds. These are Cartesian in coordinate and momentum space and chosen such that several components are identically zero for motion on the invariant manifold. In this sense these coordinates are collective. We make a connection to Zickendraht's collective coordinates and present certain configurations of few-body systems where rotations and vibrations decouple from single-particle motion. These configurations do not depend on details of the interaction.
Behzadan, A
2015-01-01
In this article we consider the conformal decomposition of Einstein's constraint equations introduced by Lichnerowicz, Choquet-Bruhat, and York, on asymptotically flat (AF) manifolds. Using the non-CMC fixed-point framework developed in 2009 by Holst, Nagy, and Tsogtgerel and by Maxwell, we establish existence of coupled non-CMC weak solutions for AF manifolds. As is the case for the analogous existence results for non-CMC solutions on closed manifolds and compact manifolds with boundary, our results here avoid the near-CMC assumption by assuming that the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small. The non-CMC rough solutions results here for AF manifolds may be viewed as extending to AF manifolds the 2009 and 2014 results on rough far-from-CMC positive Yamabe solutions for closed and compact manifolds with boundary. Similarly, our results may be viewed as extending the recent 2014 results for AF m...
On the string topology category of compact Lie groups
2013-01-01
This paper examines the string topology category of a manifold, defined by Blumberg, Cohen and Teleman. Since the string topology category is a subcategory of a compactly generated triangulated category, the machinery of stratification, constructed by Benson, Krause and Iyengar, can be applied in order to gain an understanding of the string topology category. It is shown that an appropriate stratification holds when the manifold in question is a simply connected compact Lie group. This last r...
Energy Technology Data Exchange (ETDEWEB)
Castrillon Lopez, M. [Facultad de Matematicas, Departamento de Geometria y Topologia (Spain)], E-mail: mcastri@mat.ucm.es; Gadea, P. M. [CSIC, Institute of Fundamental Physics (Spain)], E-mail: pmgadea@iec.csic.es; Oubina, J. A. [Universidade de Santiago de Compostela, Departamento de Xeometria e Topoloxia, Facultade de Matematicas (Spain)], E-mail: jaoubina@usc.es
2009-02-15
For each non-compact quaternion-Kaehler symmetric space of dimension eight, all of its descriptions as a homogeneous Riemannian space, and the associated homogeneous quaternionic Kaehler structures obtained through the Witte's refined Langlands decomposition, are studied.
Weakly asymptotically hyperbolic manifolds
Allen, Paul T; Lee, John M; Allen, Iva Stavrov
2015-01-01
We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to $-1$ and are $C^0$, but are not necessarily $C^1$, conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves as an obstruction to "higher order decay" of the Riemann curvature operator. Finally, we establish Fredholm results for geometric elliptic operators, extending the work of Rafe Mazzeo and John M. Lee to this setting. As an application, we show that any weakly asymptotically hyperbolic metric is conformally related to a weakly asymptotically hyperbolic metric of constant negative curvature.
An Explicit Nonlinear Mapping for Manifold Learning
Qiao, Hong; Zhang, Peng; Wang, Di; Zhang, Bo
2010-01-01
Manifold learning is a hot research topic in the field of computer science and has many applications in the real world. A main drawback of manifold learning methods is, however, that there is no explicit mappings from the input data manifold to the output embedding. This prohibits the application of manifold learning methods in many practical problems such as classification and target detection. Previously, in order to provide explicit mappings for manifold learning methods, many methods have...
Nonlinear manifold representations for functional data
Chen, Dong; Müller, Hans-Georg
2012-01-01
For functional data lying on an unknown nonlinear low-dimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components. Our manifold learning procedures borrow ideas from existing nonlinear dimension reduction methods, which...
Non-Riemannian effective spacetime effects on Hawking radiation in superfluids
Garcia de Andrade, L C
2005-01-01
Riemannian effective spacetime description of Hawking radiation in $^{3}He-A$ superfluids is extended to non-Riemannian effective spacetime. An example is given of non-Riemannian effective geometry of the rotational motion of the superfluid vacuum around the vortex where the effective spacetime Cartan torsion can be associated to the Hawking giving rise to a physical interpretation of effective torsion recently introduced in the literature in the form of an acoustic torsion in superfluid $^{4}He$ (PRD-70(2004),064004). Curvature and torsion singularities of this $^{3}He-A$ fermionic superfluid are investigated. This Lense-Thirring effective metric, representing the superfluid vacuum in rotational motion, is shown not support Hawking radiation when the isotropic $^{4}He$ is restored at far distances from the vortex axis. Hawking radiation can be expressed also in topological solitons (moving domain walls) in fermionic superfluids in non-Riemannian (teleparallel) $(1+1)$ dimensional effective spacetime. A telep...
On the scalar manifold of exceptional supergravity
Energy Technology Data Exchange (ETDEWEB)
Cacciatori, S.L. [Dipartimento di Scienze ed Alta Tecnologia, Universita dell' Insubria, Via Valleggio, 11, 22100 Como (Italy); INFN, Sezione di Milano, Via Celoria, 16, 20133 Milano (Italy); Cerchiai, B.L. [INFN, Sezione di Milano, Via Celoria, 16, 20133 Milano (Italy); Dipartimento di Matematica, Universita degli Studi di Milano, Via Saldini, 50, 20133 Milano (Italy); Marrani, A. [Physics Department, Theory Unit, CERN, 1211, Geneva 23 (Switzerland)
2012-07-15
We construct two parametrizations of the non compact exceptional Lie group G = E{sub 7(-25)}, based on a fibration which has the maximal compact subgroup [(E{sub 6} x U(1))/Z{sub 3}] as a fiber. It is well known that G plays an important role in the N = 2 d = 4 magic exceptional supergravity, where it describes the U-duality of the theory and where the symmetric space M=G/K gives the vector multiplets' scalar manifold. First, by making use of the exponential map, we compute a realization of G/K, that is based on the E{sub 6} invariant d-tensor, and hence exhibits the maximal possible manifest [(E{sub 6} x U(1))/Z{sub 3}]-covariance. This provides a basis for the corresponding supergravity theory, which is the analogue of the Calabi-Vesentini coordinates. Then we study the Iwasawa decomposition. Its main feature is that it is SO(8)-covariant and therefore it highlights the role of triality. Along the way we analyze the relevant chain of maximal embeddings which leads to SO(8). It is worth noticing that being based on the properties of a ''mixed'' Freudenthal-Tits magic square, the whole procedure can be generalized to a broader class of groups of type E{sub 7}. (Copyright copyright 2012 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Higher spin approaches to quantum field theory and (psuedo)-Riemannian geometries
Hallowell, Karl Evan
In this thesis, we study a number of higher spin quantum field theories and some of their algebraic and geometric consequences. These theories apply mostly either over constant curvature or more generally symmetric pseudo-Riemannian manifolds. The first part of this dissertation covers a superalgebra coming from a family of particle models over symmetric spaces. These theories are novel in that the symmetries of the (super)algebra osp( Q|2p) are larger and more elaborate than traditional symmetries. We construct useful (super)algebras related to and generalizing old work by Lichnerowicz and describe their role in developing the geometry of massless models with osp(Q|2 p) symmetry. The result is two practical applications of these (super)algebras: (1) a lunch more concise description of a family of higher spin quantum field theories; and (2) an interesting algebraic probe of underlying background geometries. We also consider massive models over constant curvature spaces. We use a radial dimensional reduction process which converts massless models into massive ones over a lower dimensional space. In our case, we take from the family of theories above the particular free, massless model over flat space associated with sp(2, R ) and derive a massive model. In the process, we develop a novel associative algebra, which is a deformation of the original differential operator algebra associated with the sp(2, R ) model. This algebra is interesting in its own right since its operators realize the representation structure of the sp(2, R ) group. The massive model also has implications for a sequence of unusual, "partially massless" theories. The derivation illuminates how reduced degrees of freedom become manifest in these particular models. Finally, we study a Yang-Mills model using an on-shell Poincare Yang-Mills twist of the Maxwell complex along with a non-minimal coupling. This is a special, higher spin case of a quantum field theory called a Yang-Mills detour complex
Flux Formulation of DFT on Group Manifolds and Generalized Scherk-Schwarz Compactifications
Bosque, Pascal du; Lust, Dieter
2015-01-01
A flux formulation of Double Field Theory on group manifold is derived and applied to study generalized Scherk-Schwarz compactifications, which give rise to a bosonic subsector of half-maximal, electrically gauged supergravities. In contrast to the flux formulation of original DFT, the covariant fluxes split into a fluctuation and a background part. The latter is connected to a $2D$-dimensional, pseudo Riemannian manifold, which is isomorphic to a Lie group embedded into O($D,D$). All fields and parameters of generalized diffeomorphisms are supported on this manifold, whose metric is spanned by the background vielbein $E_A{}^I \\in$ GL($2D$). This vielbein takes the role of the twist in conventional generalized Scherk-Schwarz compactifications. By doing so, it solves the long standing problem of constructing an appropriate twist for each solution of the embedding tensor. Using the geometric structure, absent in original DFT, $E_A{}^I$ is identified with the left invariant Maurer-Cartan form on the group manifo...
Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold
Fine, Joel
2008-01-01
Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c_1=0 which are never Kahler; e.g., we produce such manifolds with b_1=0=b_3 and also with c_2.omega <0, answering questions posed by Smith-Thomas-Yau. Examples come from Riemannian geometry, via the Levi-Civita connection on Lambda^+. The underlying six-manifold is then the twistor space and often the symplectic structure tames the Eells-Salamon twistor almost complex structure. Our second aim is to exploit this to deduce new results about minimal surfaces: if a certain curvature inequality holds, it follows that the space of minimal surfaces (with fixed topological invariants) is compactifiable; the minimal surfaces must also satisfy an adjunction in...
Fivebranes and 3-manifold homology
Gukov, Sergei; Vafa, Cumrun
2016-01-01
Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-cohomology of the Hilbert space of partially topologically twisted 3d N=2 theory T[M_3] on a Riemann surface with defects. We demonstrate this by concrete and explicit calculations in the case of monopole/Heegaard Floer homology and a 3-manifold analog of Khovanov-Rozansky link homology. The latter gives a categorification of Chern-Simons partition function. Some of the new key elements include the explicit form of the S-transform and a novel connection between categorification and a previously mysterious role of Eichler integrals in Chern-Simons theory.
Lattice QCD on nonorientable manifolds
Mages, Simon; Tóth, Bálint C.; Borsányi, Szabolcs; Fodor, Zoltán; Katz, Sándor D.; Szabó, Kálmán K.
2017-05-01
A common problem in lattice QCD simulations on the torus is the extremely long autocorrelation time of the topological charge when one approaches the continuum limit. The reason is the suppressed tunneling between topological sectors. The problem can be circumvented by replacing the torus with a different manifold, so that the connectivity of the configuration space is changed. This can be achieved by using open boundary conditions on the fields, as proposed earlier. It has the side effect of breaking translational invariance strongly. Here we propose to use a nonorientable manifold and show how to define and simulate lattice QCD on it. We demonstrate in quenched simulations that this leads to a drastic reduction of the autocorrelation time. A feature of the new proposal is that translational invariance is preserved up to exponentially small corrections. A Dirac fermion on a nonorientable manifold poses a challenge to numerical simulations: the fermion determinant becomes complex. We propose two approaches to circumvent this problem.
Motion Planning via Manifold Samples
Salzman, Oren; Raveh, Barak; Halperin, Dan
2011-01-01
We present a general and modular algorithmic framework for path planning of robots. Our framework combines geometric methods for exact and complete analysis of low-dimensional configuration spaces, together with practical, considerably simpler sampling-based approaches that are appropriate for higher dimensions. In order to facilitate the transfer of advanced geometric algorithms into practical use, we suggest taking samples that are entire low-dimensional manifolds of the configuration space that capture the connectivity of the configuration space much better than isolated point samples. Geometric algorithms for analysis of low-dimensional manifolds then provide powerful primitive operations. The modular design of the framework enables independent optimization of each modular component. Indeed, we have developed, implemented and optimized a primitive operation for complete and exact combinatorial analysis of a certain set of manifolds, using arrangements of curves of rational functions and concepts of generi...
Fivebranes and 3-manifold homology
Gukov, Sergei; Putrov, Pavel; Vafa, Cumrun
2017-07-01
Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-cohomology of the Hilbert space of partially topologically twisted 3d N=2 theory T[ M 3] on a Riemann surface with defects. We demonstrate this by concrete and explicit calculations in the case of monopole/Heegaard Floer homology and a 3-manifold analog of Khovanov-Rozansky link homology. The latter gives a categorification of Chern-Simons partition function. Some of the new key elements include the explicit form of the S-transform and a novel connection between categorification and a previously mysterious role of Eichler integrals in Chern-Simons theory.
Killing Symmetry on Finsler Manifold
Ootsuka, Takayoshi; Ishida, Muneyuki
2016-01-01
Killing vector fields $K$ are defined on Finsler manifold. The Killing symmetry is reformulated simply as $\\delta K^\\flat =0$ by using the Killing non-linear 1-form $K^\\flat$ and the spray operator $\\delta$ with the Finsler non-linear connection. $K^\\flat$ is related to the generalization of Killing tensors on Finsler manifold, and the condition $\\delta K^\\flat =0$ gives an analytical method of finding higher derivative conserved quantities, which may be called hidden conserved quantities. We show two examples: the Carter constant on Kerr spacetime and the Runge-Lentz vectors in Newtonian gravity.
Invariant manifolds and collective coordinates
Energy Technology Data Exchange (ETDEWEB)
Papenbrock, T. [Centro Internacional de Ciencias, Cuernavaca, Morelos (Mexico); Institute for Nuclear Theory, University of Washington, Seattle, WA (United States); Seligman, T.H. [Centro Internacional de Ciencias, Cuernavaca, Morelos (Mexico); Centro de Ciencias Fisicas, University of Mexico (UNAM), Cuernavaca (Mexico)
2001-09-14
We introduce suitable coordinate systems for interacting many-body systems with invariant manifolds. These are Cartesian in coordinate and momentum space and chosen such that several components are identically zero for motion on the invariant manifold. In this sense these coordinates are collective. We make a connection to Zickendraht's collective coordinates and present certain configurations of few-body systems where rotations and vibrations decouple from single-particle motion. These configurations do not depend on details of the interaction. (author)
Stein Manifolds and Holomorphic Mappings
Forstneric, Franc
2011-01-01
The main theme of this book is the homotopy principle for holomorphic mappings from Stein manifolds to the newly introduced class of Oka manifolds. This book contains the first complete account of Oka-Grauert theory and its modern extensions, initiated by Mikhail Gromov and developed in the last decade by the author and his collaborators. Included is the first systematic presentation of the theory of holomorphic automorphisms of complex Euclidean spaces, a survey on Stein neighborhoods, connections between the geometry of Stein surfaces and Seiberg-Witten theory, and a wide variety of applicat
Dynamics of Plant Growth; A Theory Based on Riemannian Geometry
Pulwicki, Julia
2016-01-01
In this work, a new model for macroscopic plant tissue growth based on dynamical Riemannian geometry is presented. We treat 1D and 2D tissues as continuous, deformable, growing geometries for sizes larger than 1mm. The dynamics of the growing tissue are described by a set of coupled tensor equations in non-Euclidean (curved) space. These coupled equations represent a novel feedback mechanism between growth and curvature dynamics. For 1D growth, numerical simulations are compared to two measures of root growth. First, modular growth along the simulated root shows an elongation zone common to many species of plant roots. Second, the relative elemental growth rate (REGR) calculated in silico exhibits temporal dynamics recently characterized in high-resolution root growth studies but which thus far lack a biological hypothesis to explain them. Namely, the REGR can evolve from a single peak localized near the root tip to a double-peak structure. In our model, this is a direct consequence of considering growth as b...
Metric measure spaces with Riemannian Ricci curvature bounded from below
Ambrosio, Luigi; Savaré, Giuseppe
2011-01-01
In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local and local-to-global properties. In these spaces, that we call RCD(K,\\infty) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the L^\\infty-Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger's relaxed...
Periodic Solutions of Lagrangian Systems on a Compact Manifold.
1983-09-01
purposes, by virtue of Theorem 3.1 it is enough to study the topology of AN. We have the following results of Vigue- Poirrier and Sullivan: Theorem 3.2...352. [VPS] Vigue- Poirrier , M. end Sullivan, D., The homology theory of the closed geodesic problem, 3. Differential Geometry 11, 633-644 (1979). VB/ed
Some Remarks on Nijenhuis Bracket, Formality, and K\\"ahler Manifolds
de Bartolomeis, Paolo
2011-01-01
One (actually, almost the only effective) way to prove formality of a differentiable manifold is to be able to produce a suitable derivation $\\delta$ such that $d\\delta$-lemma holds. We first show that such derivation $\\delta$ generates a (1,1)-tensor field (we denote it by $R$). Then, we show that the supercommutation of $d$ and $\\delta$ (which is a natural, essentially necessary condition to get a $d\\delta$-lemma) is equivalent to vanishing of the Nijenhujis torsion of $R$. Then, we are looking for sufficient conditions that ensure the $d\\delta$-lemma holds: we consider the cases when $R$ is self adjoint with respect to a Riemannian metric or compatible with an almost symplectic structure. Finally, we show that if $R$ is scew-symmetric with respect to a Riemannian metric, has constant determinant, and if its Nijenhujis torsion vanishes, then the orthogonal component of $R$ in its polar decomposition is a complex structure compatible with the metric, which gives us a new characterization of K\\"ahler structur...
An introduction to Seiberg-Witten theory on closed 3-manifolds
Bohn, Michael
2007-01-01
This is a version of the author's diploma thesis written at the University of Cologne in 2002/03. The topic is the construction of Seiberg-Witten invariants of closed 3-manifolds. In analogy to the four dimensional case, the structure of the moduli space is investigated. The Seiberg-Witten invariants are defined and their behaviour under deformation of the Riemannian metric is analyzed. Since it is essentially an exposition of results which were already known during the time of writing, the thesis has not been published. In particular, the author does not claim any originality concerning the results. Moreover, new developments of the theory are not included. However, the detailed account--together with the appendices on the required functional analytic and geometric background--might be of interest for people starting to work in the area of gauge field theory.
Spinor Fields Classification in Arbitrary Dimensions and New Classes of Spinor Fields on 7-Manifolds
Bonora, L; da Rocha, Roldao
2014-01-01
A classification of spinor fields according to the associated bilinear covariants is constructed in arbitrary dimensions and metric signatures, generalizing Lounesto's 4D spinor field classification. In such a generalized classification a basic role is played by the geometric Fierz identities. In 4D Minkowski spacetime the standard bilinear covariants can be either null or non-null -- with the exception of the current density which is invariably different from zero for physical reasons -- and sweep all types of spinor fields, including Dirac, Weyl, Majorana and more generally flagpoles, flag-dipoles and dipole spinor fields. To obtain an analogous classification in higher dimensions we use the Fierz identities, which constrain the covariant bilinears in the spinor fields and force some of them to vanish. A generalized graded Fierz aggregate is moreover obtained in such a context simply from the completeness relation. We analyze the particular and important case of Riemannian 7-manifolds, where the Majorana sp...
ON COMPLETE SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE IN NEGATIVE PINCHED MANIFOLDS
Institute of Scientific and Technical Information of China (English)
Leng Yan; Xu Hongwei
2007-01-01
A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n + p)-dimensional manifold Nn+p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H ＞ 1 there exists a negative number τ(n,p, H) ∈ (-1, 0) with the property that if the sectional curvature of N is pinched in [-1, τ(n,p, H)], and if the squared length of the second fundamental form is in a certain interval, then Nn+p is isometric to the hyperbolic space Hn+p(-1). As a consequence, this submanifold M is congruent to Sn(1/ H2-1) or theVeronese surface in S4(1/√H2-1).
Layered models for closed 3-manifolds
Johnson, Jesse
2010-01-01
We define a combinatorial structure on 3-manifolds that combines the model manifolds constructed in Minsky's proof of the ending lamination conjecture with the layered triangulations defined by Jaco and Rubinstein.
Holomorphic flexibility properties of complex manifolds
2004-01-01
We obtain results on approximation of holomorphic maps by algebraic maps, jet transversality theorems for holomorphic and algebraic maps, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds.
Nonlinear manifold representations for functional data
Chen, Dong; 10.1214/11-AOS936
2012-01-01
For functional data lying on an unknown nonlinear low-dimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components. Our manifold learning procedures borrow ideas from existing nonlinear dimension reduction methods, which we modify to address functional data settings. In simulations and applications, we study examples of functional data which lie on a manifold and validate the superior behavior of manifold mean and functional manifold components over traditional cross-sectional mean and functional principal components. We also include consistency proofs for our estimators under certain assumptions.
Periodic solutions and slow manifolds
Verhulst, F.
2006-01-01
After reviewing a number of results from geometric singular perturbation theory, we give an example of a theorem for periodic solutions in a slow manifold. This is illustrated by examples involving the van der Pol-equation and a modified logistic equation. Regarding nonhyperbolic transitions we disc
Melnikov Vector and Heteroclinic Manifolds
Institute of Scientific and Technical Information of China (English)
朱德明
1994-01-01
Using the exponential dichotomies,the transversality theory and the generalized Melnikov method,we consider the conditions for the persistence and the transversality of the singular orbit,with high degeneracy,situated on the heteroclinic or homoclinic manifold under perturbation.The results obtained extend,include and improve the corresponding ones given in certain papers well known in this area.
Cobordism Independence of Grassmann Manifolds
Indian Academy of Sciences (India)
Ashish Kumar Das
2004-02-01
This note proves that, for $F=\\mathbb{R},\\mathbb{C}$ or $\\mathbb{H}$, the bordism classes of all non-bounding Grassmannian manifolds $G_k(F^{n+k})$, with < and having real dimension , constitute a linearly independent set in the unoriented bordism group $\\mathfrak{N}_d$ regarded as a $\\mathbb{Z}_2$-vector space.
Fluid delivery manifolds and microfluidic systems
Energy Technology Data Exchange (ETDEWEB)
Renzi, Ronald F.; Sommer, Gregory J.; Singh, Anup K.; Hatch, Anson V.; Claudnic, Mark R.; Wang, Ying-Chih; Van de Vreugde, James L.
2017-02-28
Embodiments of fluid distribution manifolds, cartridges, and microfluidic systems are described herein. Fluid distribution manifolds may include an insert member and a manifold base and may define a substantially closed channel within the manifold when the insert member is press-fit into the base. Cartridges described herein may allow for simultaneous electrical and fluidic interconnection with an electrical multiplex board and may be held in place using magnetic attraction.
$\\mathcal{N}=2$ supersymmetric field theories on 3-manifolds with A-type boundaries
Aprile, Francesco
2016-01-01
General half-BPS A-type boundary conditions are formulated for N=2 supersymmetric field theories on compact 3-manifolds with boundary. We observe that under suitable conditions manifolds of the real A-type admitting two complex supersymmetries (related by charge conjugation) possess, besides a contact structure, a natural integrable toric foliation. A boundary, or a general co-dimension-1 defect, can be inserted along any leaf of this preferred foliation to produce manifolds with boundary that have the topology of a solid torus. We show that supersymmetric field theories on such manifolds can be endowed with half-BPS A-type boundary conditions. We specify the natural curved space generalization of the A-type projection of bulk supersymmetries and analyze the resulting A-type boundary conditions in generic 3d non-linear sigma models and YM/CS-matter theories.
Quantization of Presymplectic Manifolds and Circle Actions
Silva, A C; Tolman, S; Silva, Ana Canas da; Karshon, Yael; Tolman, Susan
1997-01-01
We prove several versions of "quantization commutes with reduction" for circle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin^c structure. Our theorems work whenever the quantization data and the reduction data are compatible; this condition always holds if we start from a presymplectic (in particular, symplectic) manifold.
Invariant manifolds for flows in Banach Spaces
Energy Technology Data Exchange (ETDEWEB)
Lu Kening.
1989-01-01
The author considers the existence, smoothness and exponential attractivity of global invariant manifolds for flow in Banach Spaces. He shows that every global invariant manifold can be expressed as a graph of a C{sup k} map, provided that the invariant manifolds are exponentially attractive. Applications go to the Reaction-Diffusion equation, the Kuramoto-Sivashinsky equation, and singular perturbed wave equation.
On the manifold-mapping optimization technique
Echeverria, D.; Hemker, P.W.
2006-01-01
In this paper, we study in some detail the manifold-mapping optimization technique introduced in an earlier paper. Manifold mapping aims at accelerating optimal design procedures that otherwise require many evaluations of time-expensive cost functions. We give a proof of convergence for the manifold
Modified Einstein-Gauss-Bonnet gravity: Riemann-Cartan and Pseudo-Riemannian cases
Özer, Hatice; Delice, Özgür
2016-01-01
A modified Einstein-Gauss-Bonnet gravity in four dimensions where the quadratic Gauss-Bonnet term is coupled to a scalar field is considered. The field equations of the model are obtained by variational methods by making use of the constrained-first order formalism covering both pseudo-Riemannian and non-Riemannian cases. In the pseudo-Riemannian case, the Lagrange multiplier forms, which impose the vanishing torsion constraint, are eliminated in favor of the remaining fields and the resulting metric field equations are expressed in terms of the double-dual curvature 2-form. In the non-Riemannian case with torsion, the field equations are expressed in terms of the pseudo-Riemannian quantities by a perturbative scheme valid for a weak coupling constant. It is shown that, for both cases, the model admits a maximally symmetric de-Sitter solution with nontrivial scalar field. Minimal coupling of a Dirac spinor to the Gauss-Bonnet modified gravity is also discussed briefly.
"Universal" inequalities for the eigenvalues of the biharmonic operator
Ilias, Said
2010-01-01
In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherical eigenmaps (this includes the compact homogeneous Riemannian spaces) and nally on domains of the hyperbolic space.
On the Residual Finiteness of Out(π1(M))of Certain Seifert Manifolds
Institute of Scientific and Technical Information of China (English)
R.B.J.T. Allenby; Goansu Kim; C.Y. Tang
2003-01-01
In [2], Grossman proved that the outer automorphism group of a compact orientable surface is residually finite, whence the mapping class groups of surface groups are residually finite. In this paper, we prove a similar result for Seifert manifolds with non-trivial boundary.
Shadowing effects in Newton’s law from compact extra dimensions
Kleidis, K.; Oikonomou, V. K.
2016-10-01
One problem that appears in theories with compact extra dimensions is the shadowing phenomenon, which makes it difficult to identify the topology and the geometry of the extra-dimensional manifold, if it exists. In this paper, we address this problem from the perspective of Newton’s law modifications caused by compact extra dimensions. After providing the modifications cause by some of the most phenomenologically interesting compact manifolds, we qualitatively study and compare the modifications that these manifolds cause to Newton’s law. As we demonstrate, the shadowing phenomenon persists in this approach too, and it seems that the shadowing phenomenon is grouped in “shadowing zones”. At these points, it is impossible to distinguish which compact space causes the modification of Newton’s law. However, away from the shadowing zones, the effects of the different compact manifolds are distinguishable. It is possible that these effects can be observed in the next generation of Newton’s law experiments.
Does Negative Type Characterize the Round Sphere?
DEFF Research Database (Denmark)
Kokkendorff, Simon Lyngby
2007-01-01
We discuss the measure theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact Riemannian manifold with symmetry ratio 1 must be a round sphere, was put forward in a previous paper....... We resolve this conjecture in the class of Riemannian symmetric spaces by showing, that a Riemannian manifold with symmetry ratio 1 must be of negative type and that the only compact Riemannian symmetric spaces of negative type are the round spheres....
Manifold seal structure for fuel cell stack
Collins, William P.
1988-01-01
The seal between the sides of a fuel cell stack and the gas manifolds is improved by adding a mechanical interlock between the adhesive sealing strip and the abutting surface of the manifolds. The adhesive is a material which can flow to some extent when under compression, and the mechanical interlock is formed providing small openings in the portion of the manifold which abuts the adhesive strip. When the manifolds are pressed against the adhesive strips, the latter will flow into and through the manifold openings to form buttons or ribs which mechanically interlock with the manifolds. These buttons or ribs increase the bond between the manifolds and adhesive, which previously relied solely on the adhesive nature of the adhesive.
Convexity of Spheres in a Manifold without Conjugate Points
Indian Academy of Sciences (India)
Akhil Ranjan; Hemangi Shah
2002-11-01
For a non-compact, complete and simply connected manifold without conjugate points, we prove that if the determinant of the second fundamental form of the geodesic spheres in is a radial function, then the geodesic spheres are convex. We also show that if is two or three dimensional and without conjugate points, then, at every point there exists a ray with no focal points on it relative to the initial point of the ray. The proofs use a result from the theory of vector bundles combined with the index lemma.
Jacobi Equations and Comparison Theorems for Corank 1 sub-Riemannian Structures with Symmetries
Li, Chengbo
2009-01-01
The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical bundle of moving frames and the complete system of symplectic invariants, called curvature maps, for parametrized curves in Lagrange Grassmannians satisfying very general assumptions. The structural equation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the normal form for the Jacobi equation along this extremal and the curvature maps can be seen as the "coefficients" of this normal form. In the case of a Riemannian metric there is only one curvature map and it is naturally related to the Riemannian sectional curvature. In the present paper we study the curvature maps for a sub-Riemannian structure on a corank 1 distribution having an additional transversal infinitesimal symmetry. After ...
Invariance for Single Curved Manifold
Castro, Pedro Machado Manhaes de
2012-08-01
Recently, it has been shown that, for Lambert illumination model, solely scenes composed by developable objects with a very particular albedo distribution produce an (2D) image with isolines that are (almost) invariant to light direction change. In this work, we provide and investigate a more general framework, and we show that, in general, the requirement for such in variances is quite strong, and is related to the differential geometry of the objects. More precisely, it is proved that single curved manifolds, i.e., manifolds such that at each point there is at most one principal curvature direction, produce invariant is surfaces for a certain relevant family of energy functions. In the three-dimensional case, the associated energy function corresponds to the classical Lambert illumination model with albedo. This result is also extended for finite-dimensional scenes composed by single curved objects. © 2012 IEEE.
Symmetries from the solution manifold
Aldaya, Víctor; Guerrero, Julio; Lopez-Ruiz, Francisco F.; Cossío, Francisco
2015-07-01
We face a revision of the role of symmetries of a physical system aiming at characterizing the corresponding Solution Manifold (SM) by means of Noether invariants as a preliminary step towards a proper, non-canonical, quantization. To this end, "point symmetries" of the Lagrangian are generally not enough, and we must resort to the more general concept of contact symmetries. They are defined in terms of the Poincaré-Cartan form, which allows us, in turn, to find the symplectic structure on the SM, through some sort of Hamilton-Jacobi (HJ) transformation. These basic symmetries are realized as Hamiltonian vector fields, associated with (coordinate) functions on the SM, lifted back to the Evolution Manifold through the inverse of this HJ mapping, that constitutes an inverse of the Noether Theorem. The specific examples of a particle moving on S3, at the mechanical level, and nonlinear SU(2)-sigma model in field theory are sketched.
Rigid subsets of symplectic manifolds
Entov, Michael
2007-01-01
We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the previous work of P.Albers) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.
Directory of Open Access Journals (Sweden)
Wafaa Batat
2010-02-01
Full Text Available In this note we prove that the Heisenberg group with a left-invariant pseudo-Riemannian metric admits a completely integrable totally geodesic distribution of codimension 1. This is on the contrary to the Riemannian case, as it was proved by T. Hangan.
Torsions of 3-dimensional manifolds
Wurzbacher, T
2002-01-01
From the reviews: "This is an excellent exposition about abelian Reidemeister torsions for three-manifolds." ―Zentralblatt Math "This monograph contains a wealth of information many topologists will find very handy. …Many of the new points of view pioneered by Turaev are gradually becoming mainstream and are spreading beyond the pure topology world. This monograph is a timely and very useful addition to the scientific literature." ―Mathematical Reviews
Koppelman formulas on flag manifolds
Samuelsson, Håkan
2010-01-01
We construct Koppelman formulas on manifolds of flags in $\\C^N$ for forms with values in any holomorphic line bundle as well as in the tautological vector bundles and their duals. As an application we obtain new explicit proofs of some vanishing theorems of the Bott-Borel-Weil type by solving the corresponding $\\debar$-equation. We also construct reproducing kernels for harmonic $(p,q)$-forms in the case of Grassmannians.
The Operator Manifold Formalism, 1
Ter-Kazarian, G T
1998-01-01
The suggested operator manifold formalism enables to develop an approach to the unification of the geometry and the field theory. We also elaborate the formalism of operator multimanifold yielding the multiworld geometry involving the spacetime continuum and internal worlds, where the subquarks are defined implying the Confinement and Gauge principles. This formalism in Part II (hep-th/9812182) is used to develop further the microscopic approach to some key problems of particle physics.
Coincidence classes in nonorientable manifolds
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available We study Nielsen coincidence theory for maps between manifolds of same dimension regardless of orientation. We use the definition of semi-index of a class, review the definition of defective classes, and study the occurrence of defective root classes. We prove a semi-index product formula for lifting maps and give conditions for the defective coincidence classes to be the only essential classes.
Manifolds of interconvertible pure states
Sinolecka, Magdalena M.; Zyczkowski, Karol; Kus, Marek
2001-01-01
Local orbits of a pure state of an N x N bi-partite quantum system are analyzed. We compute their dimensions which depends on the degeneracy of the vector of coefficients arising by the Schmidt decomposition. In particular, the generic orbit has 2N^2 -N-1 dimensions, the set of separable states is 4(N-1) dimensional, while the manifold of maximally entangled states has N^2-1 dimensions.
Manifolds of interconvertible pure states
Sinolecka, M M; Kus, M; Sinolecka, Magdalena M.; Zyczkowski, Karol; Kus, Marek
2002-01-01
Local orbits of a pure state of an N x N bi-partite quantum system are analyzed. We compute their dimensions which depends on the degeneracy of the vector of coefficients arising by the Schmidt decomposition. In particular, the generic orbit has 2N^2 -N-1 dimensions, the set of separable states is 4(N-1) dimensional, while the manifold of maximally entangled states has N^2-1 dimensions.
Model Reduction by Manifold Boundaries
Transtrum, Mark K.; Qiu, Peng
2015-01-01
Understanding the collective behavior of complex systems from their basic components is a difficult yet fundamental problem in science. Existing model reduction techniques are either applicable under limited circumstances or produce “black boxes” disconnected from the microscopic physics. We propose a new approach by translating the model reduction problem for an arbitrary statistical model into a geometric problem of constructing a low-dimensional, submanifold approximation to a high-dimensional manifold. When models are overly complex, we use the observation that the model manifold is bounded with a hierarchy of widths and propose using the boundaries as submanifold approximations. We refer to this approach as the manifold boundary approximation method. We apply this method to several models, including a sum of exponentials, a dynamical systems model of protein signaling, and a generalized Ising model. By focusing on parameters rather than physical degrees of freedom, the approach unifies many other model reduction techniques, such as singular limits, equilibrium approximations, and the renormalization group, while expanding the domain of tractable models. The method produces a series of approximations that decrease the complexity of the model and reveal how microscopic parameters are systematically “compressed” into a few macroscopic degrees of freedom, effectively building a bridge between the microscopic and the macroscopic descriptions. PMID:25216014
Non-Riemannian geometry: towards new avenues for the physics of modified gravity
Olmo, Gonzalo J
2015-01-01
Less explored than their metric (Riemannian) counterparts, metric-affine (or Palatini) theories bring an unexpected phenomenology for gravitational physics beyond General Relativity. Lessons of crystalline structures, where the presence of defects in their microstructure requires the use of non-Riemannian geometry for the proper description of their properties in the macroscopic continuum level, are discussed. In this analogy, concepts such as wormholes and geons play a fundamental role. Applications of the metric-affine formalism developed by the authors in the last three years are reviewed.
Double Field Theory on Group Manifolds (Thesis)
Hassler, Falk
2015-01-01
This thesis deals with Double Field Theory (DFT), an effective field theory capturing the low energy dynamics of closed strings on a torus. It renders T-duality on a torus manifest by adding $D$ winding coordinates in addition to the $D$ space time coordinates. An essential consistency constraint of the theory, the strong constraint, only allows for field configurations which depend on half of the coordinates of the arising doubled space. I derive DFT${}_\\mathrm{WZW}$, a generalization of the current formalism. It captures the low energy dynamics of a closed bosonic string propagating on a compact group manifold. Its classical action and the corresponding gauge transformations arise from Closed String Field Theory up to cubic order in the massless fields. These results are rewritten in terms of a generalized metric and extended to all orders in the fields. There is an explicit distinction between background and fluctuations. For the gauge algebra to close, the latter have to fulfill a modified strong constrai...
Constructing reference metrics on multicube representations of arbitrary manifolds
Lindblom, Lee; Taylor, Nicholas W.; Rinne, Oliver
2016-05-01
Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous. This paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies. The method is tested here by constructing reference metrics for compact, orientable two-dimensional manifolds with genera between zero and five. These metrics are shown to satisfy the Gauss-Bonnet identity numerically to the level of truncation error (which converges toward zero as the numerical resolution is increased). These reference metrics can be made smoother and more uniform by evolving them with Ricci flow. This smoothing procedure is tested on the two-dimensional reference metrics constructed here. These smoothing evolutions (using volume-normalized Ricci flow with DeTurck gauge fixing) are all shown to produce reference metrics with constant scalar curvatures (at the level of numerical truncation error).
Compactness and gluing theory for monopoles
Frøyshov, Kim A
2008-01-01
This book is devoted to the study of moduli spaces of Seiberg-Witten monopoles over spinc Riemannian 4–manifolds with long necks and/or tubular ends. The original purpose of this work was to provide analytical foundations for a certain construction of Floer homology of rational homology 3–spheres; this is carried out in [Monopole Floer homology for rational homology 3–spheres arXiv:08094842]. However, along the way the project grew, and, except for some of the transversality results, most of the theory is developed more generally than is needed for that construction. Floer homology itself is hardly touched upon in this book, and, to compensate for that, I have included another application of the analytical machinery, namely a proof of a "generalized blow-up formula" which is an important tool for computing Seiberg–Witten invariants. The book is divided into three parts. Part 1 is almost identical to my paper [Monopoles over 4–manifolds containing long necks I, Geom. Topol. 9 (2005) 1–93]. The oth...
Infinitesimal moduli of G2 holonomy manifolds with instanton bundles
de la Ossa, Xenia; Svanes, Eirik Eik
2016-01-01
We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomology $H^*_{{\\check{\\rm d}}_E}(Y,E)$ developed by Reyes-Carri\\'on and Fern\\'andez and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{{\\check{\\rm d}}_A}(Y,{\\rm End}(V))$ plus the moduli of the $G_2$ structure preserving the instanton condition. The latter piece is contained in $H^1_{{\\check{\\rm d}}_\
The Multidimensional Darboux Transformation
González-López, A; González-López, Artemio; Kamran, Niky
1996-01-01
A generalization of the classical one-dimensional Darboux transformation to arbitrary n-dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical two-dimensional Moutard transformation is also generalized to non-compact oriented Riemannian manifolds of dimension n greater than one. New examples of quasi-exactly solvable multidimensional matrix Schrödinger operators on curved manifolds are obtained by applying the above results.
Partial Regularity for Higher Dimensional Landau-Lifshitz Systems
Institute of Scientific and Technical Information of China (English)
丁时进; 郭柏灵
2003-01-01
@@ 1 Initial Value Problem Let M be a m-dimensional compact, smooth, Riemannian manifold without boundary. For simplicity, we shall assume M = Rm in our proof since the general case can be handled in the similar manner.
Smooth Maps of a Foliated Manifold in a Symplectic Manifold
Indian Academy of Sciences (India)
Mahuya Datta; Md Rabiul Islam
2009-06-01
Let be a smooth manifold with a regular foliation $\\mathcal{F}$ and a 2-form which induces closed forms on the leaves of $\\mathcal{F}$ in the leaf topology. A smooth map $f:(M,\\mathcal{F})\\longrightarrow(N, )$ in a symplectic manifold $(N, )$ is called a foliated symplectic immersion if restricts to an immersion on each leaf of the foliation and further, the restriction of $f^∗$ is the same as the restriction of on each leaf of the foliation. If is a foliated symplectic immersion then the derivative map $Df$ gives rise to a bundle morphism $F:TM\\longrightarrow TN$ which restricts to a monomorphism on $T\\mathcal{F}\\subseteq TM$ and satisfies the condition $F^∗=$ on $T\\mathcal{F}$. A natural question is whether the existence of such a bundle map ensures the existence of a foliated symplectic immersion . As we shall see in this paper, the obstruction to the existence of such an is only topological in nature. The result is proved using the ℎ-principle theory of Gromov.
A Note on the Foucault Pendulum and the Sub-Riemannian Formalism
Anzaldo-Meneses, A.; Monroy-Pérez, F.
The well known Foucault pendulum is studied within the formalism of sub-Riemannian geometry on a step-2 nilpotent Lie group. For small oscillations, trajectories are explicitly calculated, they turn out to be hypotrochöids obtained by rolling without slipping a circle onto another circle.
Differential Calculus on N-Graded Manifolds
Sardanashvily, G.; W. Wachowski
2017-01-01
The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, over N-graded commutative rings and on N-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and on Z2-graded manifolds. We follow the notion of an N-graded manifold as a local-ringed space whose body is a s...
Discrete equations and the singular manifold method
Estévez, P G
1999-01-01
The Painleve expansion for the second Painleve equation (PII) and fourth Painleve equation (PIV) have two branches. The singular manifold method therefore requires two singular manifolds. The double singular manifold method is used to derive Miura transformations from PII and PIV to modified Painleve type equations for which auto-Backlund transformations are obtained. These auto-Backlund transformations can be used to obtain discrete equations.
OBJECTORIENTED NUMERICAL MANIFOLD METHOD
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
The design and management of the objects about the numerical manifold method are studied by abstracting the finite cover system of numerical manifold method as independent data classes and the theoretical basis for the researching and expanding of numerical manifold method is also put forward. The Hammer integration of triangular area coordinates is used in the integration of the element. The calculation result shows that the program is accuracy and effective.
Homology group on manifolds and their foldings
Directory of Open Access Journals (Sweden)
M. Abu-Saleem
2010-03-01
Full Text Available In this paper, we introduce the definition of the induced unfolding on the homology group. Some types of conditional foldings restricted on the elements of the homology groups are deduced. The effect of retraction on the homology group of a manifold is dicussed. The unfolding of variation curvature of manifolds on their homology group are represented. The relations between homology group of the manifold and its folding are deduced.
Similarity Learning of Manifold Data.
Chen, Si-Bao; Ding, Chris H Q; Luo, Bin
2015-09-01
Without constructing adjacency graph for neighborhood, we propose a method to learn similarity among sample points of manifold in Laplacian embedding (LE) based on adding constraints of linear reconstruction and least absolute shrinkage and selection operator type minimization. Two algorithms and corresponding analyses are presented to learn similarity for mix-signed and nonnegative data respectively. The similarity learning method is further extended to kernel spaces. The experiments on both synthetic and real world benchmark data sets demonstrate that the proposed LE with new similarity has better visualization and achieves higher accuracy in classification.
Hidden torsion, 3-manifolds, and homology cobordism
Cha, Jae Choon
2011-01-01
This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov rho-invariants associated to amenable representations. We introduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraic approximation we call local hidden torsion. We construct infinitely many hyperbolic 3-manifolds which have local hidden torsion in the transfinite lower central subgroup. By realizing Cheeger-Gromov invariants over amenable groups, we show that our hyperbolic 3-manifolds are not pairwise homology cobordant, yet remain indistinguishable by any prior known homology cobordism invariants.
Differential Calculus on N-Graded Manifolds
Directory of Open Access Journals (Sweden)
G. Sardanashvily
2017-01-01
Full Text Available The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, over N-graded commutative rings and on N-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and on Z2-graded manifolds. We follow the notion of an N-graded manifold as a local-ringed space whose body is a smooth manifold Z. A key point is that the graded derivation module of the structure ring of graded functions on an N-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its body Z. Accordingly, the Chevalley–Eilenberg differential calculus on an N-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus on N-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds of N-graded bundles.
The Fibered Isomorphism Conjecture for Complex Manifolds
Institute of Scientific and Technical Information of China (English)
S. K. ROUSHON
2007-01-01
In this paper we show that the Fibered Isomorphism Conjecture of Farrell and Jones,corresponding to the stable topological pseudoisotopy functor, is true for the fundamental groups of a class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative K-groups of the fundamental groups of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds including a large class of 3-manifolds which has a finite sheeted cover fibering over the circle.
Manifold knowledge extraction and target recognition
Chao, Cai; Hua, Zhou
2009-10-01
Advanced mammalian target identification derived from the perception of target's manifold and measurement manifolddistance. It does not rely on object's segmented accuracy, not depend on target's variety model, and adapt to a range of changes on targets. In this paper, based on the existed manifold learning algorithm, set up a new bionic automatic target recognition model, discussed the targets manifold knowledge acquisition and the knowledge expression architecture, gave a manifold knowledge-based new method for automatic target recognition. Experiments show that the new method has a strong adaptability to targets various transform, and has a very high correctly identification probability.
Spectral gaps, inertial manifolds and kinematic dynamos
Energy Technology Data Exchange (ETDEWEB)
Nunez, Manuel [Departamento de Analisis Matematico, Universidad de Valladolid, 47005 Valladolid (Spain)]. E-mail: mnjmhd@am.uva.es
2005-10-17
Inertial manifolds are desirable objects when ones wishes a dynamical process to behave asymptotically as a finite-dimensional ones. Recently [Physica D 194 (2004) 297] these manifolds are constructed for the kinematic dynamo problem with time-periodic velocity. It turns out, however, that the conditions imposed on the fluid velocity to guarantee the existence of inertial manifolds are too demanding, in the sense that they imply that all the solutions tend exponentially to zero. The inertial manifolds are meaningful because they represent different decay rates, but the classical dynamos where the magnetic field is maintained or grows are not covered by this approach, at least until more refined estimates are found.
Wilce, A
2004-01-01
We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and also the projection lattice of a Hilbert space. As the latter example illustrates, a lattice-ordered TOA need not be a topological lattice. However, we show that a compact Boolean TOA is a topological Boolean algebra. Using this, we prove that any compact regular TOA is atomistic, and has a compact center. We prove also that any compact TOA with isolated 0 is of finite height. We then focus on stably ordered TOAs: those in which the upper-set generated by an open set is open. These include both topological orthomodular lattices and interval orthoalgebras -- in particular, projection lattices. We show that the topology of a compact stably-ordered TOA with isolated 0 is determined by that of of its space of atoms.
Manifold learning in protein interactomes.
Marras, Elisabetta; Travaglione, Antonella; Capobianco, Enrico
2011-01-01
Many studies and applications in the post-genomic era have been devoted to analyze complex biological systems by computational inference methods. We propose to apply manifold learning methods to protein-protein interaction networks (PPIN). Despite their popularity in data-intensive applications, these methods have received limited attention in the context of biological networks. We show that there is both utility and unexplored potential in adopting manifold learning for network inference purposes. In particular, the following advantages are highlighted: (a) fusion with diagnostic statistical tools designed to assign significance to protein interactions based on pre-selected topological features; (b) dissection into components of the interactome in order to elucidate global and local connectivity organization; (c) relevance of embedding the interactome in reduced dimensions for biological validation purposes. We have compared the performances of three well-known techniques--kernel-PCA, RADICAL ICA, and ISOMAP--relatively to their power of mapping the interactome onto new coordinate dimensions where important associations among proteins can be detected, and then back projected such that the corresponding sub-interactomes are reconstructed. This recovery has been done selectively, by using significant information according to a robust statistical procedure, and then standard biological annotation has been provided to validate the results. We expect that a byproduct of using subspace analysis by the proposed techniques is a possible calibration of interactome modularity studies. Supplementary Material is available online at www.libertonlinec.com.
Directory of Open Access Journals (Sweden)
Jun Zhang
2013-12-01
Full Text Available Divergence functions are the non-symmetric “distance” on the manifold, Μθ, of parametric probability density functions over a measure space, (Χ,μ. Classical information geometry prescribes, on Μθ: (i a Riemannian metric given by the Fisher information; (ii a pair of dual connections (giving rise to the family of α-connections that preserve the metric under parallel transport by their joint actions; and (iii a family of divergence functions ( α-divergence defined on Μθ x Μθ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μθ (that of parametric probability density functions to the manifold, Μ, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M can be interpreted as arising from an embedding of Μθ into Μ. Furthermore, when a parametric model (after a monotonic scaling forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α = ± 1, generalizing the results of Amari’s α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function expressed in the divergence function (“referential duality” and the other concerning its representation under an arbitrary monotone scaling (“representational duality”.
Integrability conditions on Engel-type manifolds
Calin, Ovidiu; Chang, Der-Chen; Hu, Jishan
2015-09-01
We introduce the concept of Engel manifold, as a manifold that resembles locally the Engel group, and find the integrability conditions of the associated sub-elliptic system , . These are given by , . Then an explicit construction of the solution involving an integral representation is provided, which corresponds to a Poincaré-type lemma for the Engel's distribution.
Target manifold formation using a quadratic SDF
Hester, Charles F.; Risko, Kelly K. D.
2013-05-01
Synthetic Discriminant Function (SDF) formulation of correlation filters provides constraints for forming target subspaces for a target set. In this paper we extend the SDF formulation to include quadratic constraints and use this solution to form nonlinear manifolds in the target space. The theory for forming these manifolds will be developed and demonstrated with data.
Einstein Constraints on Asymptotically Euclidean Manifolds
Choquet-Bruhat, Y; York, J W; Choquet-Bruhat, Yvonne; Isenberg, James; York, James W.
2000-01-01
We consider the Einstein constraints on asymptotically euclidean manifolds $M$ of dimension $n \\geq 3$ with sources of both scaled and unscaled types. We extend to asymptotically euclidean manifolds the constructive method of proof of existence. We also treat discontinuous scaled sources. In the last section we obtain new results in the case of non-constant mean curvature.
Li, Tao
2011-01-01
We construct a counterexample to the Rank versus Genus Conjecture, i.e. a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus. Moreover, we show that the discrepancy between rank and Heegaard genus can be arbitrarily large for hyperbolic 3-manifolds. We also construct toroidal such examples containing hyperbolic JSJ pieces.
An Explicit Nonlinear Mapping for Manifold Learning.
Qiao, Hong; Zhang, Peng; Wang, Di; Zhang, Bo
2013-02-01
Manifold learning is a hot research topic in the held of computer science and has many applications in the real world. A main drawback of manifold learning methods is, however, that there are no explicit mappings from the input data manifold to the output embedding. This prohibits the application of manifold learning methods in many practical problems such as classification and target detection. Previously, in order to provide explicit mappings for manifold learning methods, many methods have been proposed to get an approximate explicit representation mapping with the assumption that there exists a linear projection between the high-dimensional data samples and their low-dimensional embedding. However, this linearity assumption may be too restrictive. In this paper, an explicit nonlinear mapping is proposed for manifold learning, based on the assumption that there exists a polynomial mapping between the high-dimensional data samples and their low-dimensional representations. As far as we know, this is the hrst time that an explicit nonlinear mapping for manifold learning is given. In particular, we apply this to the method of locally linear embedding and derive an explicit nonlinear manifold learning algorithm, which is named neighborhood preserving polynomial embedding. Experimental results on both synthetic and real-world data show that the proposed mapping is much more effective in preserving the local neighborhood information and the nonlinear geometry of the high-dimensional data samples than previous work.
Gauged supergravities from Bianchi's group manifolds
Bergshoeff, E; Gran, U; Linares, R; Nielsen, M; Ortin, T; Roest, D
2004-01-01
We construct maximal D = 8 gauged supergravities by the reduction of D = I I supergravity over three-dimensional group manifolds. Such manifolds are classified into two classes, A and B, and eleven types. This Bianchi classification carries over to the gauged supergravities. The class A theories hav
Simplicial approach to derived differential manifolds
Borisov, Dennis
2011-01-01
Derived differential manifolds are constructed using the usual homotopy theory of simplicial rings of smooth functions. They are proved to be equivalent to derived differential manifolds of finite type, constructed using homotopy sheaves of homotopy rings (D.Spivak), thus preserving the classical cobordism ring. This reduction to the usual algebraic homotopy can potentially lead to virtual fundamental classes beyond obstruction theory.
Strictly convex functions on complete Finsler manifolds
Indian Academy of Sciences (India)
YOE ITOKAWA; KATSUHIRO SHIOHAMA; BANKTESHWAR TIWARI
2016-10-01
The purpose of the present paper is to investigate the influence of strictly convex functions on the metric structures of complete Finsler manifolds. More precisely we discuss the properties of the group of isometries and the exponential maps on a complete Finsler manifold admitting strictly convex functions.
Souza, Fábio S
2011-01-01
We give examples of open smooth manifolds that cannot be leaves of any Riemannian foliation of arbitrary codimension on a compact manifold. We also present a new class of non-leaves of C^0 codimension one foliations, simply connected manifolds of dimension at least 5 that are non-periodic in homotopy, namely in their 2-dimensional homotopy groups.
Manifold-based learning and synthesis.
Huang, Dong; Yi, Zhang; Pu, Xiaorong
2009-06-01
This paper proposes a new approach to analyze high-dimensional data set using low-dimensional manifold. This manifold-based approach provides a unified formulation for both learning from and synthesis back to the input space. The manifold learning method desires to solve two problems in many existing algorithms. The first problem is the local manifold distortion caused by the cost averaging of the global cost optimization during the manifold learning. The second problem results from the unit variance constraint generally used in those spectral embedding methods where global metric information is lost. For the out-of-sample data points, the proposed approach gives simple solutions to transverse between the input space and the feature space. In addition, this method can be used to estimate the underlying dimension and is robust to the number of neighbors. Experiments on both low-dimensional data and real image data are performed to illustrate the theory.
Heterotic model building: 16 special manifolds
Energy Technology Data Exchange (ETDEWEB)
He, Yang-Hui [Department of Mathematics, City University,London, EC1V 0HB (United Kingdom); School of Physics, NanKai University,Tianjin, 300071 (China); Merton College, University of Oxford,Oxford OX14JD (United Kingdom); Lee, Seung-Joo [School of Physics, Korea Institute for Advanced Study,Seoul 130-722 (Korea, Republic of); Lukas, Andre; Sun, Chuang [Rudolf Peierls Centre for Theoretical Physics, University of Oxford,1 Keble Road, Oxford OX1 3NP (United Kingdom)
2014-06-12
We study heterotic model building on 16 specific Calabi-Yau manifolds constructed as hypersurfaces in toric four-folds. These 16 manifolds are the only ones among the more than half a billion manifolds in the Kreuzer-Skarke list with a non-trivial first fundamental group. We classify the line bundle models on these manifolds, both for SU(5) and SO(10) GUTs, which lead to consistent supersymmetric string vacua and have three chiral families. A total of about 29000 models is found, most of them corresponding to SO(10) GUTs. These models constitute a starting point for detailed heterotic model building on Calabi-Yau manifolds in the Kreuzer-Skarke list. The data for these models can be downloaded http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/toricdata/index.html.
Heisenberg symmetry and hypermultiplet manifolds
Antoniadis, Ignatios; Petropoulos, P Marios; Siampos, Konstantinos
2015-01-01
We study the emergence of Heisenberg (Bianchi II) algebra in hyper-K\\"ahler and quaternionic spaces. This is motivated by the r\\^ole these spaces with this symmetry play in $\\mathcal{N}=2$ hypermultiplet scalar manifolds. We show how to construct related pairs of hyper-K\\"ahler and quaternionic spaces under general symmetry assumptions, the former being a zooming-in limit of the latter at vanishing cosmological constant. We further apply this method for the two hyper-K\\"ahler spaces with Heisenberg algebra, which is reduced to $U(1)\\times U(1)$ at the quaternionic level. We also show that no quaternionic spaces exist with a strict Heisenberg symmetry -- as opposed to $\\text{Heisenberg} \\ltimes U(1)$. We finally discuss the realization of the latter by gauging appropriate $Sp(2,4)$ generators in $\\mathcal{N}=2$ conformal supergravity.
Moving Manifolds in Electromagnetic Fields
Directory of Open Access Journals (Sweden)
David V. Svintradze
2017-08-01
Full Text Available We propose dynamic non-linear equations for moving surfaces in an electromagnetic field. The field is induced by a material body with a boundary of the surface. Correspondingly the potential energy, set by the field at the boundary can be written as an addition of four-potential times four-current to a contraction of the electromagnetic tensor. Proper application of the minimal action principle to the system Lagrangian yields dynamic non-linear equations for moving three dimensional manifolds in electromagnetic fields. The equations in different conditions simplify to Maxwell equations for massless three surfaces, to Euler equations for a dynamic fluid, to magneto-hydrodynamic equations and to the Poisson-Boltzmann equation.
Heisenberg symmetry and hypermultiplet manifolds
Directory of Open Access Journals (Sweden)
Ignatios Antoniadis
2016-04-01
Full Text Available We study the emergence of Heisenberg (Bianchi II algebra in hyper-Kähler and quaternionic spaces. This is motivated by the rôle these spaces with this symmetry play in N=2 hypermultiplet scalar manifolds. We show how to construct related pairs of hyper-Kähler and quaternionic spaces under general symmetry assumptions, the former being a zooming-in limit of the latter at vanishing scalar curvature. We further apply this method for the two hyper-Kähler spaces with Heisenberg algebra, which is reduced to U(1×U(1 at the quaternionic level. We also show that no quaternionic spaces exist with a strict Heisenberg symmetry – as opposed to Heisenberg⋉U(1. We finally discuss the realization of the latter by gauging appropriate Sp(2,4 generators in N=2 conformal supergravity.
Function theory on symplectic manifolds
Polterovich, Leonid
2014-01-01
This is a book on symplectic topology, a rapidly developing field of mathematics which originated as a geometric tool for problems of classical mechanics. Since the 1980s, powerful methods such as Gromov's pseudo-holomorphic curves and Morse-Floer theory on loop spaces gave rise to the discovery of unexpected symplectic phenomena. The present book focuses on function spaces associated with a symplectic manifold. A number of recent advances show that these spaces exhibit intriguing properties and structures, giving rise to an alternative intuition and new tools in symplectic topology. The book provides an essentially self-contained introduction into these developments along with applications to symplectic topology, algebra and geometry of symplectomorphism groups, Hamiltonian dynamics and quantum mechanics. It will appeal to researchers and students from the graduate level onwards. I like the spirit of this book. It formulates concepts clearly and explains the relationship between them. The subject matter is i...
Harmonic space and quaternionic manifolds
Galperin, A; Ogievetsky, O V
1994-01-01
We find a principle of harmonic analyticity underlying the quaternionic (quaternion-K\\"ahler) geometry and solve the differential constraints which define this geometry. To this end the original $4n$-dimensional quaternionic manifold is extended to a bi-harmonic space. The latter includes additional harmonic coordinates associated with both the tangent local $Sp(1)$ group and an extra rigid $SU(2)$ group rotating the complex structures. Then the constraints can be rewritten as integrability conditions for the existence of an analytic subspace in the bi-harmonic space and solved in terms of two unconstrained potentials on the analytic subspace. Geometrically, the potentials have the meaning of vielbeins associated with the harmonic coordinates. We also establish a one-to-one correspondence between the quaternionic spaces and off-shell $N=2$ supersymmetric sigma-models coupled to $N=2$ supergravity. The general $N=2$ sigma-model Lagrangian when written in the harmonic superspace is composed of the quaternionic ...
Manifold Matching for High-Dimensional Pattern Recognition
HOTTA, Seiji
2008-01-01
In this chapter manifold matching for high-dimensional pattern classification was described. The topics described in this chapter were summarized as follows: The meaning and effectiveness of manifold matching The similarity between various classifiers from the point of view of manifold matching Accuracy improvement for manifold matching Learning rules for manifold matching Experimental results on handwritten digit datasets showed that manifold matching achieved lower error rates than other cl...
Line bundle twisted chiral de Rham complex and bound states of D-branes on toric manifolds
Energy Technology Data Exchange (ETDEWEB)
Parkhomenko, S.E., E-mail: spark@itp.ac.ru [Landau Institute for Theoretical Physics, 142432 Chernogolovka, Moscow region (Russian Federation); Moscow Institute of Physics and Technology, 141707 Dolgoprudny, Moscow region (Russian Federation)
2014-04-15
In this note we calculate elliptic genus in various examples of twisted chiral de Rham complex on two-dimensional toric compact manifolds and Calabi–Yau hypersurfaces in toric manifolds. At first the elliptic genus is calculated for the line bundle twisted chiral de Rham complex on a compact smooth toric manifold and K3 hypersurface in P{sup 3}. Then we twist chiral de Rham complex by sheaves localized on positive codimension submanifolds in P{sup 2} and calculate in each case the elliptic genus. In the last example the elliptic genus of chiral de Rham complex on P{sup 2} twisted by SL(N) vector bundle with instanton number k is calculated. In all the cases considered we find the infinite tower of open string oscillator contributions and identify directly the open string boundary conditions of the corresponding bound state of D-branes.
Butail, Sachit; Bollt, Erik M; Porfiri, Maurizio
2013-11-07
In this paper, we build a framework for the analysis and classification of collective behavior using methods from generative modeling and nonlinear manifold learning. We represent an animal group with a set of finite-sized particles and vary known features of the group structure and motion via a class of generative models to position each particle on a two-dimensional plane. Particle positions are then mapped onto training images that are processed to emphasize the features of interest and match attainable far-field videos of real animal groups. The training images serve as templates of recognizable patterns of collective behavior and are compactly represented in a low-dimensional space called embedding manifold. Two mappings from the manifold are derived: the manifold-to-image mapping serves to reconstruct new and unseen images of the group and the manifold-to-feature mapping allows frame-by-frame classification of raw video. We validate the combined framework on datasets of growing level of complexity. Specifically, we classify artificial images from the generative model, interacting self-propelled particle model, and raw overhead videos of schooling fish obtained from the literature. © 2013 Elsevier Ltd. All rights reserved.
Conformal invariance of massless Duffin-Kemmer-Petiau theory in Riemannian spacetimes
Energy Technology Data Exchange (ETDEWEB)
Casana, R [Instituto de Fisica Teorica, Universidade Estadual Paulista, Rua Pamplona 145, CEP 01405-900, Sao Paulo, SP (Brazil); Lunardi, J T [Grupo de Fisica Teorica, Departamento de Matematica e Estatistica, Universidade Estadual de Ponta Grossa, Av. Gal. Carlos Cavalcanti 4748, 84032-900, Ponta Grossa, PR (Brazil); Pimentel, B M [Instituto de Fisica Teorica, Universidade Estadual Paulista, Rua Pamplona 145, CEP 01405-900, Sao Paulo, SP (Brazil); Teixeira, R G [Departamento de Fisica, Universidade Federal do EspIrito Santo, Av. Fernando Ferrari s/n, Goiabeiras, CEP 29060-900, Vitoria, ES (Brazil)
2005-07-21
We investigate the conformal invariance of massless Duffin-Kemmer-Petiau theory coupled to Riemannian spacetimes. We show that, as usual, in the minimal coupling procedure only the spin 1 sector of the theory-which corresponds to the electromagnetic field-is conformally invariant. We also show that the conformal invariance of the spin 0 sector can be naturally achieved by introducing a compensating term in the Lagrangian. Such a procedure-besides not modifying the spin 1 sector-leads to the well-known conformal coupling between the scalar curvature and the massless Klein-Gordon-Fock field. Going beyond the Riemannian spacetimes, we briefly discuss the effects of a nonvanishing torsion in the scalar case.
Conformal invariance of massless Duffin Kemmer Petiau theory in Riemannian spacetimes
Casana, R.; Lunardi, J. T.; Pimentel, B. M.; Teixeira, R. G.
2005-07-01
We investigate the conformal invariance of massless Duffin Kemmer Petiau theory coupled to Riemannian spacetimes. We show that, as usual, in the minimal coupling procedure only the spin 1 sector of the theory—which corresponds to the electromagnetic field—is conformally invariant. We also show that the conformal invariance of the spin 0 sector can be naturally achieved by introducing a compensating term in the Lagrangian. Such a procedure—besides not modifying the spin 1 sector—leads to the well-known conformal coupling between the scalar curvature and the massless Klein Gordon Fock field. Going beyond the Riemannian spacetimes, we briefly discuss the effects of a nonvanishing torsion in the scalar case.
The existence of Hamiltonian stationary Lagrangian tori in Kahler manifolds of any dimension
Lee, Yng-Ing
2010-01-01
Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They can be considered as a generalization of special Lagrangians or Lagrangian and minimal submanifolds. Joyce, Schoen and the author show that given any compact rigid Hamiltonian stationary Lagrangian in $\\C^n$, one can always find a family of Hamiltonian stationary Lagrangians of the same type in any compact symplectic manifolds with a compatible metric. The advantage of this result is that it holds in very general classes. But the disadvantage is that we do not know where these examples locate and examples in this family might be far apart. In this paper, we derive a local condition on Kahler manifolds which ensures the existence of one family of Hamiltonian stationary Lagrangian tori near a point with given frame satisfying the criterion. Butscher and Corvino ever proposed a condition in n=2. But our condition appears to be different from theirs. The condition d...
Discriminative sparse coding on multi-manifolds
Wang, J.J.-Y.
2013-09-26
Sparse coding has been popularly used as an effective data representation method in various applications, such as computer vision, medical imaging and bioinformatics. However, the conventional sparse coding algorithms and their manifold-regularized variants (graph sparse coding and Laplacian sparse coding), learn codebooks and codes in an unsupervised manner and neglect class information that is available in the training set. To address this problem, we propose a novel discriminative sparse coding method based on multi-manifolds, that learns discriminative class-conditioned codebooks and sparse codes from both data feature spaces and class labels. First, the entire training set is partitioned into multiple manifolds according to the class labels. Then, we formulate the sparse coding as a manifold-manifold matching problem and learn class-conditioned codebooks and codes to maximize the manifold margins of different classes. Lastly, we present a data sample-manifold matching-based strategy to classify the unlabeled data samples. Experimental results on somatic mutations identification and breast tumor classification based on ultrasonic images demonstrate the efficacy of the proposed data representation and classification approach. 2013 The Authors. All rights reserved.
DEFF Research Database (Denmark)
Bjerre, Troels; Hansen, Mads Fogtmann; Aznar, M.;
2012-01-01
For deformable registration of computed tomography (CT) scans in image guided radiation therapy (IGRT) we apply Riemannian elasticity regularization. We explore the use of spatially varying elasticity parameters to encourage bone rigidity and local tissue volume change only in the gross tumor......-model we achieved a total mean target registration error (TRE) of 0.92 ± 0.49 mm. Using spatially varying regularization for the HL case, deformation was limited to the GTV and lungs....
Differential rotation of stretched and twisted thick magnetic flux tube dynamos in Riemannian spaces
de Andrade, Garcia
2007-01-01
The topological mapping between a torus of big radius and a sphere is applied to the Riemannian geometry of a stretched and twisted very thick magnetic flux tube, to obtain spherical dynamos solving the magnetohydrodynamics (MHD) self-induction equation for the magnetic flux tubes undergoing differential (non-uniform) rotation along the tube magnetic axis. Constraints on the shear is also computed. It is shown that when the hypothesis of the convective cyclonic dynamo is used the rotation is ...
Space time manifolds and contact structures
Directory of Open Access Journals (Sweden)
K. L. Duggal
1990-01-01
Full Text Available A new class of contact manifolds (carring a global non-vanishing timelike vector field is introduced to establish a relation between spacetime manifolds and contact structures. We show that odd dimensional strongly causal (in particular, globally hyperbolic spacetimes can carry a regular contact structure. As examples, we present a causal spacetime with a non regular contact structure and a physical model [Gödel Universe] of Homogeneous contact manifold. Finally, we construct a model of 4-dimensional spacetime of general relativity as a contact CR-submanifold.
Higher Order Hessian Structures on Manifolds
Indian Academy of Sciences (India)
R David Kumar
2005-08-01
In this paper we define th order Hessian structures on manifolds and study them. In particular, when =3, we make a detailed study and establish a one-to-one correspondence between third-order Hessian structures and a certain class of connections on the second-order tangent bundle of a manifold. Further, we show that a connection on the tangent bundle of a manifold induces a connection on the second-order tangent bundle. Also we define second-order geodesics of special second-order connection which gives a geometric characterization of symmetric third-order Hessian structures.
Loops in Reeb Graphs of 2-Manifolds
Energy Technology Data Exchange (ETDEWEB)
Cole-McLaughlin, K; Edelsbrunner, H; Harer, J; Natarajan, V; Pascucci, V
2004-12-16
Given a Morse function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(n log n), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse function.
Loops in Reeb Graphs of 2-Manifolds
Energy Technology Data Exchange (ETDEWEB)
Cole-McLaughlin, K; Edelsbrunner, H; Harer, J; Natarajan, V; Pascucci, V
2003-02-11
Given a Morse function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(n log n), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse function.
Mean-field cosmological dynamos in Riemannian space with isotropic diffusion
de Andrade, L Garcia
2009-01-01
Mean-field cosmological dynamos in Riemannian space with isotropic diffusion}} Previous attempts for building a cosmic dynamo including preheating in inflationary universes [Bassett et al Phys Rev (2001)] has not included mean field or turbulent dynamos. In this paper a mean field dynamo in cosmic scales on a Riemannian spatial cosmological section background, is set up. When magnetic fields and flow velocities are parallel propagated along the Riemannian space dynamo action is obtained. Turbulent diffusivity ${\\beta}$ is coupled with the Ricci magnetic curvature, as in Marklund and Clarkson [MNRAS (2005)], GR-MHD dynamo equation. Mean electric field possesses an extra term where Ricci tensor couples with magnetic vector potential in Ohm's law. In Goedel universe induces a mean field dynamo growth rate ${\\gamma}=2{\\omega}^{2}{\\beta}$. In this frame kinetic helicity vanishes. In radiation era this yields ${\\gamma}\\approx{2{\\beta}{\\times}10^{-12}s^{-1}}$. In non-comoving the magnetic field is expressed as $B\\ap...
Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps
Pfaff, Jonathan
2012-01-01
For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of SL_2(C) to the corresponding Reidemeister torsion. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti.
THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C(1) smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.
On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds
Directory of Open Access Journals (Sweden)
Kazuhiko Fukui
2017-01-01
Full Text Available We proved in [K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. 54 (1978, 52-54] that the identity component \\(\\text{Diff}\\,^r_{G,c}(M_0\\ of the group of equivariant \\(C^r\\-diffeomorphisms of a principal \\(G\\ bundle \\(M\\ over a manifold \\(B\\ is perfect for a compact connected Lie group \\(G\\ and \\(1 \\leq r \\leq \\infty\\ (\\(r \
NMS Flows on Three-Dimensional Manifolds with One Saddle Periodic Orbit
Institute of Scientific and Technical Information of China (English)
B. CAMPOS; A. CORDERO; J. Mart(i)nez ALFARO; P. VINDEL
2004-01-01
The simplest NMS flow is a polar flow formed by an attractive periodic orbit and a repulsive periodic orbit as limit sets. In this paper we show that the only orientable, simple, compact,3-dimensional manifolds without boundary that admit an NMS flow with none or one saddle periodic orbit are lens spaces.We also see that when a fattened round handle is a connected sum of tori, the corresponding flow is also a trivial connected sum of flows.
Gicquaud, Romain
2014-01-01
We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3-manifolds admitting a $\\bS^1$-symmetry group. When the quotient manifold has genus greater than 2, we obtain strong far from CMC results.
Crystal Melting and Toric Calabi-Yau Manifolds
Ooguri, Hirosi
2008-01-01
We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau threefold. The three-dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary non-compact toric Calabi-Yau manifold. We point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the Donaldson-Thomas theory.
Analysis III analytic and differential functions, manifolds and Riemann surfaces
Godement, Roger
2015-01-01
Volume III sets out classical Cauchy theory. It is much more geared towards its innumerable applications than towards a more or less complete theory of analytic functions. Cauchy-type curvilinear integrals are then shown to generalize to any number of real variables (differential forms, Stokes-type formulas). The fundamentals of the theory of manifolds are then presented, mainly to provide the reader with a "canonical'' language and with some important theorems (change of variables in integration, differential equations). A final chapter shows how these theorems can be used to construct the compact Riemann surface of an algebraic function, a subject that is rarely addressed in the general literature though it only requires elementary techniques. Besides the Lebesgue integral, Volume IV will set out a piece of specialized mathematics towards which the entire content of the previous volumes will converge: Jacobi, Riemann, Dedekind series and infinite products, elliptic functions, classical theory of modular fun...
Symplectic manifolds with no Kähler structure
Tralle, Aleksy
1997-01-01
This is a research monograph covering the majority of known results on the problem of constructing compact symplectic manifolds with no Kaehler structure with an emphasis on the use of rational homotopy theory. In recent years, some new and stimulating conjectures and problems have been formulated due to an influx of homotopical ideas. Examples include the Lupton-Oprea conjecture, the Benson-Gordon conjecture, both of which are in the spirit of some older and still unsolved problems (e.g. Thurston's conjecture and Sullivan's problem). Our explicit aim is to clarify the interrelations between certain aspects of symplectic geometry and homotopy theory in the framework of the problems mentioned above. We expect that the reader is aware of the basics of differential geometry and algebraic topology at graduate level.
Branched standard spines of 3-manifolds
Benedetti, Riccardo
1997-01-01
This book provides a unified combinatorial realization of the categroies of (closed, oriented) 3-manifolds, combed 3-manifolds, framed 3-manifolds and spin 3-manifolds. In all four cases the objects of the realization are finite enhanced graphs, and only finitely many local moves have to be taken into account. These realizations are based on the notion of branched standard spine, introduced in the book as a combination of the notion of branched surface with that of standard spine. The book is intended for readers interested in low-dimensional topology, and some familiarity with the basics is assumed. A list of questions, some of which concerning relations with the theory of quantum invariants, is enclosed.
3-manifold groups are virtually residually p
Aschenbrenner, Matthias
2010-01-01
Given a prime $p$, a group is called residually $p$ if the intersection of its $p$-power index normal subgroups is trivial. A group is called virtually residually $p$ if it has a finite index subgroup which is residually $p$. It is well-known that finitely generated linear groups over fields of characteristic zero are virtually residually $p$ for all but finitely many $p$. In particular, fundamental groups of hyperbolic $3$-manifolds are virtually residually $p$. It is also well-known that fundamental groups of $3$-manifolds are residually finite. In this paper we prove a common generalization of these results: every $3$-manifold group is virtually residually $p$ for all but finitely many~$p$. This gives evidence for the conjecture (Thurston) that fundamental groups of $3$-manifolds are linear groups.
Hierarchical manifold learning for regional image analysis.
Bhatia, Kanwal K; Rao, Anil; Price, Anthony N; Wolz, Robin; Hajnal, Joseph V; Rueckert, Daniel
2014-02-01
We present a novel method of hierarchical manifold learning which aims to automatically discover regional properties of image datasets. While traditional manifold learning methods have become widely used for dimensionality reduction in medical imaging, they suffer from only being able to consider whole images as single data points. We extend conventional techniques by additionally examining local variations, in order to produce spatially-varying manifold embeddings that characterize a given dataset. This involves constructing manifolds in a hierarchy of image patches of increasing granularity, while ensuring consistency between hierarchy levels. We demonstrate the utility of our method in two very different settings: 1) to learn the regional correlations in motion within a sequence of time-resolved MR images of the thoracic cavity; 2) to find discriminative regions of 3-D brain MR images associated with neurodegenerative disease.
Regional manifold learning for disease classification.
Ye, Dong Hye; Desjardins, Benoit; Hamm, Jihun; Litt, Harold; Pohl, Kilian M
2014-06-01
While manifold learning from images itself has become widely used in medical image analysis, the accuracy of existing implementations suffers from viewing each image as a single data point. To address this issue, we parcellate images into regions and then separately learn the manifold for each region. We use the regional manifolds as low-dimensional descriptors of high-dimensional morphological image features, which are then fed into a classifier to identify regions affected by disease. We produce a single ensemble decision for each scan by the weighted combination of these regional classification results. Each weight is determined by the regional accuracy of detecting the disease. When applied to cardiac magnetic resonance imaging of 50 normal controls and 50 patients with reconstructive surgery of Tetralogy of Fallot, our method achieves significantly better classification accuracy than approaches learning a single manifold across the entire image domain.
Particle Filtering on the Audio Localization Manifold
Ettinger, Evan
2010-01-01
We present a novel particle filtering algorithm for tracking a moving sound source using a microphone array. If there are N microphones in the array, we track all $N \\choose 2$ delays with a single particle filter over time. Since it is known that tracking in high dimensions is rife with difficulties, we instead integrate into our particle filter a model of the low dimensional manifold that these delays lie on. Our manifold model is based off of work on modeling low dimensional manifolds via random projection trees [1]. In addition, we also introduce a new weighting scheme to our particle filtering algorithm based on recent advancements in online learning. We show that our novel TDOA tracking algorithm that integrates a manifold model can greatly outperform standard particle filters on this audio tracking task.
Polynomial chaos representation of databases on manifolds
Energy Technology Data Exchange (ETDEWEB)
Soize, C., E-mail: christian.soize@univ-paris-est.fr [Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-La-Vallée, Cedex 2 (France); Ghanem, R., E-mail: ghanem@usc.edu [University of Southern California, 210 KAP Hall, Los Angeles, CA 90089 (United States)
2017-04-15
Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.
Mathematical Background of Formalism of Operator Manifold
Ter-Kazarian, G T
1997-01-01
The analysis of mathematical structure of the method of operator manifold guides our discussion. The latter is a still wider generalization of the method of secondary quantization with appropriate expansion over the geometric objects. The nature of operator manifold provides its elements with both quantum field and geometry aspects, a detailed study of which is a subject of present paper. It yields a quantization of geometry differing in principle from all earlier suggested schemes.
Multiply manifolded molten carbonate fuel cells
Energy Technology Data Exchange (ETDEWEB)
Krumpelt, M.; Roche, M.F.; Geyer, H.K.; Johnson, S.A.
1994-08-01
This study consists of research and development activities related to the concept of a molten carbonate fuel cell (MCFC) with multiple manifolds. Objective is to develop an MCFC having a higher power density and a longer life than other MCFC designs. The higher power density will result from thinner gas flow channels; the extended life will result from reduced temperature gradients. Simplification of the gas flow channels and current collectors may also significantly reduce cost for the multiply manifolded MCFC.
Blowing up generalized Kahler 4-manifolds
Cavalcanti, Gil R
2011-01-01
We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a blow-up operation for bi-Hermitian manifolds.
On some applications of invariant manifolds
Institute of Scientific and Technical Information of China (English)
Xi-Yun Hou; Lin Liu; Yu-Hui Zhao
2011-01-01
Taking transfer orbits of a collinear libration point probe, a lunar probe and an interplanetary probe as examples, some applications of stable and unstable invariant manifolds of the restricted three-body problem are discussed. Research shows that transfer energy is not necessarily conserved when invariant manifolds are used. For the cases in which the transfer energy is conserved, the cost is a much longer transfer time.
Infinitesimal moduli of G2 holonomy manifolds with instanton bundles
de la Ossa, Xenia; Larfors, Magdalena; Svanes, Eirik E.
2016-11-01
We describe the infinitesimal moduli space of pairs ( Y, V) where Y is a manifold with G 2 holonomy, and V is a vector bundle on Y with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical G 2 cohomology developed by Reyes-Carrión and Fernández and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli {H}_{{overset{ěe }{d}}_A}^1(Y,End(V)) plus the moduli of the G 2 structure preserving the instanton condition. The latter piece is contained in {H}_{overset{ěe }{d}θ}^1(Y,TY) , and is given by the kernel of a map overset{ěe }{F} which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the map overset{ěe }{F} is given in terms of the curvature of the bundle and maps {H}_{overset{ěe }{d}θ}^1(Y,TY) into {H}_{{overset{ěe }{d}}_A}^2(Y,End(V)) , and moreover can be used to define a cohomology on an extension bundle of TY by End( V). We comment further on the resemblance with the holomorphic Atiyah algebroid and connect the story to physics, in particular to heterotic compactifications on ( Y, V) when α' = 0.
Quaternionic-like manifolds and homogeneous twistor spaces.
Pantilie, Radu
2016-12-01
Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the 'quaternionic-like manifolds'. These contain, as particular subclasses, the CR quaternionic and the ρ-quaternionic manifolds. Moreover, the notion of 'heaven space' finds its adequate level of generality in this setting: (essentially) any real analytic quaternionic-like manifold admits a (germ) unique heaven space, which is a ρ-quaternionic manifold. We, also, give a natural construction of homogeneous complex manifolds endowed with embedded spheres, thus, emphasizing the abundance of the quaternionic-like manifolds.
Robinson manifolds and Cauchy-Riemann spaces
Trautman, A
2002-01-01
A Robinson manifold is defined as a Lorentz manifold (M, g) of dimension 2n >= 4 with a bundle N subset of C centre dot TM such that the fibres of N are maximal totally null and there holds the integrability condition [Sec N, Sec N] subset of Sec N. The real part of N intersection N-bar is a bundle of null directions tangent to a congruence of null geodesics. This generalizes the notion of a shear-free congruence of null geodesics (SNG) in dimension 4. Under a natural regularity assumption, the set M of all these geodesics has the structure of a Cauchy-Riemann manifold of dimension 2n - 1. Conversely, every such CR manifold lifts to many Robinson manifolds. Three definitions of a CR manifold are described here in considerable detail; they are equivalent under the assumption of real analyticity, but not in the smooth category. The distinctions between these definitions have a bearing on the validity of the Robinson theorem on the existence of null Maxwell fields associated with SNGs. This paper is largely a re...
On the Picard group of a compact flat projective variety
Michelacakis, NJ
1996-01-01
In this note, we describe the Picard group of the class of compact, smooth, flat, projective varieties. In view of Charlap's work and Johnson's characterization, we construct line bundles over such manifolds as the holonomy-invariant elements of the Neron-Severi group of a projective flat torus cove
一类对偶平坦的黎曼度量%A Class of Dually Flat Riemannian Metrics
Institute of Scientific and Technical Information of China (English)
田艳芳; 杨秀文; 林琼; 徐维
2014-01-01
In this paper, a sufficient condition of locally dual flat in Riemannian space is obtained:an equation that the spray of a Riemannian metric satisfies. At the same time, the theory what this condition is not necessary is pointed out since an example is given to prove. Further research is finished to characterize the quality of this kind of Riemannian metrics. The equivalent condition that this kind of locally dually flat Riemannian metric is Einstein metrics is disussed. The quality of this kind of locally dually flat Riemannian metric is been researched to show that they are Einstein metrics. Here Riemannian curvature is main consideration. A series of computation shows that a locally dually flat Riemannian metric is Einstein metric if and only if it is Euclidian with dimen-sion n≥3 . But this is not suitable for the space with dimension n=2.%给出了黎曼度量局部对偶平坦的一个充分条件：黎曼度量的Spray所满足的方程。同时，指出该条件是非必要的，并给出了相关反例。进一步，对满足条件的这类黎曼度量的性质进行了研究。具体地，讨论了这类度量成为Einstein度量的条件。从黎曼曲率着手，通过计算发现：当空间维数n≥3，这类黎曼度量是Einstein度量，当且仅当它是欧氏度量；但是，这个结论对n=2的情形不适用。
Viewpoint Manifolds for Action Recognition
Directory of Open Access Journals (Sweden)
Souvenir Richard
2009-01-01
Full Text Available Abstract Action recognition from video is a problem that has many important applications to human motion analysis. In real-world settings, the viewpoint of the camera cannot always be fixed relative to the subject, so view-invariant action recognition methods are needed. Previous view-invariant methods use multiple cameras in both the training and testing phases of action recognition or require storing many examples of a single action from multiple viewpoints. In this paper, we present a framework for learning a compact representation of primitive actions (e.g., walk, punch, kick, sit that can be used for video obtained from a single camera for simultaneous action recognition and viewpoint estimation. Using our method, which models the low-dimensional structure of these actions relative to viewpoint, we show recognition rates on a publicly available dataset previously only achieved using multiple simultaneous views.
Viewpoint Manifolds for Action Recognition
Directory of Open Access Journals (Sweden)
Richard Souvenir
2009-01-01
Full Text Available Action recognition from video is a problem that has many important applications to human motion analysis. In real-world settings, the viewpoint of the camera cannot always be fixed relative to the subject, so view-invariant action recognition methods are needed. Previous view-invariant methods use multiple cameras in both the training and testing phases of action recognition or require storing many examples of a single action from multiple viewpoints. In this paper, we present a framework for learning a compact representation of primitive actions (e.g., walk, punch, kick, sit that can be used for video obtained from a single camera for simultaneous action recognition and viewpoint estimation. Using our method, which models the low-dimensional structure of these actions relative to viewpoint, we show recognition rates on a publicly available dataset previously only achieved using multiple simultaneous views.
Control of nonholonomic systems from sub-Riemannian geometry to motion planning
Jean, Frédéric
2014-01-01
Nonholonomic systems are control systems which depend linearly on the control. Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. In particular the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The aim of these notes is to present these notions of approximation and their application to the motion planning problem for nonholonomic systems.
Energy Technology Data Exchange (ETDEWEB)
Bluemich, Bernhard; Haber-Pohlmeier, Sabina; Zia, Wasif [RWTH Aachen Univ. (Germany). Inst. fuer Technische und Makromolekulare Chemie (ITMC)
2014-06-01
Nuclear Magnetic Resonance (NMR) spectroscopy is the most popular method for chemists to analyze molecular structures, while Magnetic Resonance Imaging (MRI) is a non-invasive diagnostic tool for medical doctors that provides high-contrast images of biological tissue. In both applications, the sample (or patient) is positioned inside a large, superconducting magnet to magnetize the atomic nuclei. Interrogating radio-frequency pulses result in frequency spectra that provide the chemist with molecular information, the medical doctor with anatomic images, and materials scientist with NMR relaxation parameters. Recent advances in magnet technology have led to a variety of small permanent magnets to allow compact and low-cost instruments. The goal of this book is to provide an introduction to the practical use of compact NMR at a level nearly as basic as the operation of a smart phone.
Bazeia, D; Marques, M A; Menezes, R; Zafalan, I
2016-01-01
We study a family of Maxwell-Higgs models, described by the inclusion of a function of the scalar field that represent generalized magnetic permeability. We search for vortex configurations which obey first-order differential equations that solve the equations of motion. We first deal with the asymptotic behavior of the field configurations, and then implement a numerical study of the solutions, the energy density and the magnetic field. We work with the generalized permeability having distinct profiles, giving rise to new models, and we investigate how the vortices behave, compared with the solutions of the corresponding standard models. In particular, we show how to build compact vortices, that is, vortex solutions with the energy density and magnetic field vanishing outside a compact region of the plane.
Energy Technology Data Exchange (ETDEWEB)
Bazeia, D.; Losano, L.; Marques, M.A.; Zafalan, I. [Universidade Federal da Paraiba, Departamento de Fisica, Joao Pessoa, PB (Brazil); Menezes, R. [Universidade Federal da Paraiba, Departamento de Ciencias Exatas, Rio Tinto, PB (Brazil); Universidade Federal de Campina Grande, Departamento de Fisica, Campina Grande, PB (Brazil)
2017-02-15
We study a family of Maxwell-Higgs models, described by the inclusion of a function of the scalar field that represent generalized magnetic permeability. We search for vortex configurations which obey first-order differential equations that solve the equations of motion. We first deal with the asymptotic behavior of the field configurations, and then implement a numerical study of the solutions, the energy density and the magnetic field. We work with the generalized permeability having distinct profiles, giving rise to new models, and we investigate how the vortices behave, compared with the solutions of the corresponding standard models. In particular, we show how to build compact vortices, that is, vortex solutions with the energy density and magnetic field vanishing outside a compact region of the plane. (orig.)
Descriptor Learning via Supervised Manifold Regularization for Multioutput Regression.
Zhen, Xiantong; Yu, Mengyang; Islam, Ali; Bhaduri, Mousumi; Chan, Ian; Li, Shuo
2016-06-08
Multioutput regression has recently shown great ability to solve challenging problems in both computer vision and medical image analysis. However, due to the huge image variability and ambiguity, it is fundamentally challenging to handle the highly complex input-target relationship of multioutput regression, especially with indiscriminate high-dimensional representations. In this paper, we propose a novel supervised descriptor learning (SDL) algorithm for multioutput regression, which can establish discriminative and compact feature representations to improve the multivariate estimation performance. The SDL is formulated as generalized low-rank approximations of matrices with a supervised manifold regularization. The SDL is able to simultaneously extract discriminative features closely related to multivariate targets and remove irrelevant and redundant information by transforming raw features into a new low-dimensional space aligned to targets. The achieved discriminative while compact descriptor largely reduces the variability and ambiguity for multioutput regression, which enables more accurate and efficient multivariate estimation. We conduct extensive evaluation of the proposed SDL on both synthetic data and real-world multioutput regression tasks for both computer vision and medical image analysis. Experimental results have shown that the proposed SDL can achieve high multivariate estimation accuracy on all tasks and largely outperforms the algorithms in the state of the arts. Our method establishes a novel SDL framework for multioutput regression, which can be widely used to boost the performance in different applications.
Knot Optimization for Biharmonic B-splines on Manifold Triangle Meshes.
Hou, Fei; He, Ying; Qin, Hong; Hao, Aimin
2017-09-01
Biharmonic B-splines, proposed by Feng and Warren, are an elegant generalization of univariate B-splines to planar and curved domains with fully irregular knot configuration. Despite the theoretic breakthrough, certain technical difficulties are imperative, including the necessity of Voronoi tessellation, the lack of analytical formulation of bases on general manifolds, expensive basis re-computation during knot refinement/removal, being applicable for simple domains only (e.g., such as euclidean planes, spherical and cylindrical domains, and tori). To ameliorate, this paper articulates a new biharmonic B-spline computing paradigm with a simple formulation. We prove that biharmonic B-splines have an equivalent representation, which is solely based on a linear combination of Green's functions of the bi-Laplacian operator. Consequently, without explicitly computing their bases, biharmonic B-splines can bypass the Voronoi partitioning and the discretization of bi-Laplacian, enable the computational utilities on any compact 2-manifold. The new representation also facilitates optimization-driven knot selection for constructing biharmonic B-splines on manifold triangle meshes. We develop algorithms for spline evaluation, data interpolation and hierarchical data decomposition. Our results demonstrate that biharmonic B-splines, as a new type of spline functions with theoretic and application appeal, afford progressive update of fully irregular knots, free of singularity, without the need of explicit parameterization, making it ideal for a host of graphics tasks on manifolds.
Characterization of a Banach-Finsler manifold in terms of the algebras of smooth functions
Jaramillo, J A; Sanchez-Gonzalez, L
2011-01-01
In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete $C^k$ Finsler manifold $M$ is determined by the normed algebra $C_b^k(M)$ of all real-valued, bounded and $C^k$ smooth functions with bounded derivative defined on $M$. As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete $C^k$ Finsler manifold $M$ is determined by the algebra $C_b^k(M)$; (ii) the weak Finsler structure of a separable and complete $C^k$ Finsler manifold $M$ modeled on a Banach space with a Lipschitz and $C^k$ smooth bump function is determined by the algebra $C^k_b(M)$; (iii) the weak Finsler structure of a $C^k$ uniformly bumpable and complete $C^k$ Finsler manifold $M$ modeled on a Weakly Compactly Generated (WCG) Banach space with an (equivalent) $C^k$ smooth norm is determined by the algebra $C^k_b(M)$; and (iii) the isometric structure of a WCG Banach space $X$ with an $C^1$ smooth bump function is determined by the algebra $C_b^1(X)$.
Lattice QCD on Non-Orientable Manifolds
Mages, Simon; Borsanyi, Szabolcs; Fodor, Zoltan; Katz, Sandor; Szabo, Kalman K
2015-01-01
A common problem in lattice QCD simulations on the torus is the extremely long autocorrelation time of the topological charge, when one approaches the continuum limit. The reason is the suppressed tunneling between topological sectors. The problem can be circumvented by replacing the torus with a different manifold, so that the field configuration space becomes connected. This can be achieved by using open boundary conditions on the fields, as proposed earlier. It has the side effect of breaking translational invariance completely. Here we propose to use a non-orientable manifold, and show how to define and simulate lattice QCD on it. We demonstrate in quenched simulations that this leads to a drastic reduction of the autocorrelation time. A feature of the new proposal is, that translational invariance is preserved up to exponentially small corrections. A Dirac-fermion on a non-orientable manifold poses a challenge to numerical simulations: the fermion determinant becomes complex. We propose two approaches to...
New Spinor Fields on Lorentzian 7-Manifolds
Bonora, L
2016-01-01
This paper deals with the classification of spinor fields according to the bilinear covariants in 7 dimensions. It extends to higher dimensions the so-called Lounesto spinor fields classification in Minkowski spacetime, which encompasses Dirac, Weyl, Majorana, and more generally flagpoles, flag-dipoles and dipole spinor fields. A generalized classification according to the bilinear covariants was previously studied on Euclidean 7-manifolds. It presents either just one spinor field class, in the real case of Majorana spinors, or three non-trivial classes in the most general case. In this paper we show that by imposing appropriate conditions on spinor fields in 7d manifolds with Lorentzian metric, the formerly obtained obstructions for new classes of spinor fields can be circumvented. New spinor fields classes are then explicitly constructed. In particular, on 7-manifolds with asymptotically flat black hole background, by means of such spinors one can introduce a generalized current density which further serves...
Unknotting tunnels in hyperbolic 3-manifolds
Adams, Colin
2012-01-01
An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic in a hyperbolic 3-manifold M, we find sufficient conditions for it to be an unknotting tunnel. In particular, if the vertical geodesic corresponds to a 4-bracelet, 5-bracelet or 6-bracelet in the universal cover and has short enough length, it must be an unknotting tunnel. Furthermore, we consider a vertical geodesic that satisfies the elder sibling property, which means that in the universal cover, every horoball except the one centered at infinity is connected to a larger horoball by a lift of the vertical geodesic. Such a vertical geodesic with length less than ln(2) is then shown to be an unknotting tunnel.
The Framework, Causal and Co-compact Structure of Space-time
Kovár, Martin
2013-01-01
We introduce a canonical, compact topology, which we call weakly causal, naturally generated by the causal site of J. D. Christensen and L. Crane, a pointless algebraic structure motivated by certain problems of quantum gravity. We show that for every four-dimensional globally hyperbolic Lorentzian manifold there exists an associated causal site, whose weakly causal topology is co-compact with respect to the manifold topology and vice versa. Thus, the causal site has the full information about the topology of space-time, represented by the Lorentzian manifold. In addition, we show that there exist also non-Lorentzian causal sites (whose causal relation is not a continuous poset) and so the weakly causal topology and its de Groot dual extends the usual manifold topology of space-time beyond topologies generated by the traditional, smooth model. As a source of inspiration in topologizing the studied causal structures, we use some methods and constructions of general topology and formal concept analysis.
Burning invariant manifolds in reactive front propagation
Mahoney, John; Mitchell, Kevin; Solomon, Tom
2011-01-01
We present theory and experiments on the dynamics of reaction fronts in a two-dimensional flow composed of a chain of alternating vortices. Inspired by the organization of passive transport by invariant manifolds, we introduce burning invariant manifolds (BIMs), which act as one-sided barriers to front propagation. The BIMs emerge from the theory when the advection-reaction- diffusion system is recast as an ODE for reaction front elements. Experimentally, we demonstrate how these BIMs can be measured and compare their behavior with simulation. Finally, a topological BIM formalism yields a maximum front propagation speed.
The "Parity" Anomaly On An Unorientable Manifold
Witten, Edward
2016-01-01
The "parity" anomaly -- more accurately described as an anomaly in time-reversal or reflection symmetry -- arises in certain theories of fermions coupled to gauge fields and/or gravity in a spacetime of odd dimension. The "parity" anomaly has traditionally been studied on orientable manifolds only, but recent developments involving topological superconductors have made it clear that one can get more information by asking what happens on an unorientable manifold. In this paper, we analyze the "parity" anomaly for fermions coupled to gauge fields and gravity in $2+1$ dimensions. We consider applications to gapped boundary states of a topological superconductor and to M2-branes in string/M-theory.
Wilson Fermions on a Randomly Triangulated Manifold
Burda, Z; Krzywicki, A
1999-01-01
A general method of constructing the Dirac operator for a randomly triangulated manifold is proposed. The fermion field and the spin connection live, respectively, on the nodes and on the links of the corresponding dual graph. The construction is carried out explicitly in 2-d, on an arbitrary orientable manifold without boundary. It can be easily converted into a computer code. The equivalence, on a sphere, of Majorana fermions and Ising spins in 2-d is rederived. The method can, in principle, be extended to higher dimensions.
Unraveling flow patterns through nonlinear manifold learning.
Tauro, Flavia; Grimaldi, Salvatore; Porfiri, Maurizio
2014-01-01
From climatology to biofluidics, the characterization of complex flows relies on computationally expensive kinematic and kinetic measurements. In addition, such big data are difficult to handle in real time, thereby hampering advancements in the area of flow control and distributed sensing. Here, we propose a novel framework for unsupervised characterization of flow patterns through nonlinear manifold learning. Specifically, we apply the isometric feature mapping (Isomap) to experimental video data of the wake past a circular cylinder from steady to turbulent flows. Without direct velocity measurements, we show that manifold topology is intrinsically related to flow regime and that Isomap global coordinates can unravel salient flow features.
Unraveling flow patterns through nonlinear manifold learning.
Directory of Open Access Journals (Sweden)
Flavia Tauro
Full Text Available From climatology to biofluidics, the characterization of complex flows relies on computationally expensive kinematic and kinetic measurements. In addition, such big data are difficult to handle in real time, thereby hampering advancements in the area of flow control and distributed sensing. Here, we propose a novel framework for unsupervised characterization of flow patterns through nonlinear manifold learning. Specifically, we apply the isometric feature mapping (Isomap to experimental video data of the wake past a circular cylinder from steady to turbulent flows. Without direct velocity measurements, we show that manifold topology is intrinsically related to flow regime and that Isomap global coordinates can unravel salient flow features.
Inertial manifold of the atmospheric equations
Institute of Scientific and Technical Information of China (English)
李建平; 丑纪范
1999-01-01
For a class of nonlinear evolution equations, their global attractors are studied and the existence of their inertial manifolds is discussed using the truncated method. Then, on the basis of the properties of operators of the atmospheric equations, it is proved that the operator equation of the atmospheric motion with dissipation and external forcing belongs to the class of nonlinear evolution equations. Therefore, it is known that there exists an inertial manifold of the atmospheric equations if the spectral gap condition for the dissipation operator is satisfied. These results furnish a basis for further studying the dynamical properties of global attractor of the atmospheric equations and for designing better numerical scheme.
Tangent bundles of Hantzsche-Wendt manifolds
Gaşior, A.; Szczepański, A.
2013-08-01
We formulate a condition for the existence of a SpinC-structure on an oriented flat manifold Mn with H2(Mn,R)=0. We prove that Mn has a SpinC-structure if and only if there exists a homomorphism ɛ:π1(Mn)→SpinC(n) such that λ∘ɛ=h, where h:π1(Mn)→SO(n) is a holonomy homomorphism and λ:SpinC(n)→SO(n) is a standard homomorphism defined. As an application we shall prove that all cyclic Hantzsche-Wendt manifolds do not have the SpinC-structure.
Beyond Sentiment: The Manifold of Human Emotions
Kim, Seungyeon; Lebanon, Guy; Essa, Irfan
2012-01-01
Sentiment analysis predicts the presence of positive or negative emotions in a text document. In this paper we consider higher dimensional extensions of the sentiment concept, which represent a richer set of human emotions. Our approach goes beyond previous work in that our model contains a continuous manifold rather than a finite set of human emotions. We investigate the resulting model, compare it to psychological observations, and explore its predictive capabilities. Besides obtaining significant improvements over a baseline without manifold, we are also able to visualize different notions of positive sentiment in different domains.
Generalized nonuniform dichotomies and local stable manifolds
Bento, António J G
2010-01-01
We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform dichotomy for the evolution operator that contains the nonuniform exponential and polynomial dichotomies as a very particular case. The family of dichotomies considered allow situations for which the classical Lyapunov exponents are zero. Additionally, we give new examples of application of our stable manifold theorem and study the behavior of the dynamics under perturbations.
Spinor formalism and the geometry of six-dimensional Riemannian spaces
Andreev, K V
2012-01-01
The article consists of the Russian and English variants of Ph.D. Thesis in which the answers is given on the following questions: 1. how to construct the spinor formalism for n=6; 2. how to construct the spinor formalism for n=8; 3. how to prolong the Riemannian connection from the tangent bundle into the spinor one with the base: a complex analytical 6-dimensional Riemannian space; 4. how to construct the real and complex representations of this bundles; 5. how to construct the curvature spinors and to investigate its properties; 6. how to obtain the canonical form of a bilinear form for the 6-dimensional pseudo-Euclidean space with the even index of the metric; 7. how to construct the geometric interpretation of isotropic twistors on the isotropic cone of the 6-dimensional pseudo-Euclidean space with the index equal to 4; 8. how to construct the generalization of the Cartan triality principle to the Klein correspondence; 9. how to construct the structural constants of the octonion algebra for the initial i...
On the embedding of Weyl manifolds
Avalos, R.; Dahia, F.; Romero, C.
2017-01-01
We discuss the possibility of extending different versions of the Campbell-Magaard theorem, which have already been established in the context of semi-Riemannian geometry, to the context of Weyl's geometry. We show that some of the known results can be naturally extended to the new geometric scenario, although new difficulties arise. In pursuit of solving the embedding problem, we have obtained some no-go theorems. We also highlight some of the difficulties that appear in the embedding problem, which are typical of the Weylian character of the geometry. The establishing of these new results may be viewed as part of a program that highlights the possible significance of embedding theorems of increasing degrees of generality in the context of modern higher-dimensional space-time theories.
A Lower Bound of the Genus of a Self-amalgamated 3-manifolds
Institute of Scientific and Technical Information of China (English)
LI XU; LEI FENG-CHUN
2011-01-01
Let M be a compact connected oriented 3-manifold with boundary,Q1, Q2 (C)(э)M be two disjoint homeomorphic subsurfaces of (э)M, and h: Q1 → Q2 be an orientation-reversing homeomorphism. Denote by Mh or MQ1＝Q2 the 3-manifold obtained from M by gluing Q1 and Q2 together via h. Mh is called a self-amalgamation of M along Q1 and Q2. Suppose Q1 and Q2 lie on the same component F′ of (э)M′, and F′ - Q1 ∪ Q2 is connected. We give a lower bound to the Heegaard genus of M when M′ has a Heegaard splitting with sufficiently high distance.
A sufficient condition for the genus of an amalgamated 3-manifold not to go down
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let M i be a connected, compact, orientable 3-manifold, F i a boundary component of M i with g(F i ) 2, i = 1, 2, and F 1 ≌ F 2 . Let : F 1 → F 2 be a homeomorphism, and M = M 1 ∪ M 2 , F = F 2 = (F 1 ). Then it is known that g(M ) g(M 1 ) + g(M 2 ) - g(F ). In the present paper, we give a sufficient condition for the genus of an amalgamated 3-manifold not to go down as follows: Suppose that there is no essential surface with boundary (Q i1 , Q i ) in (M i1 , F i ) satisfying χ(Q i ) > 3 - 2g(M i ), i = 1, 2. Then g(M ) = g(M 1 ) + g(M 2 ) - g(F ).
$\\rm G_2$ holonomy manifolds are superconformal
Díaz, Lázaro O Rodríguez
2016-01-01
We study the chiral de Rham complex (CDR) over a manifold $M$ with holonomy $\\rm G_2$. We prove that the vertex algebra of global sections of the CDR associated to $M$ contains two commuting copies of the Shatashvili-Vafa $\\rm G_2$ superconformal algebra. Our proof is a tour de force, based on explicit computations.
Four-manifolds, geometries and knots
Hillman, Jonathan A
2007-01-01
The goal of this book is to characterize algebraically the closed 4-manifolds that fibre nontrivially or admit geometries in the sense of Thurston, or which are obtained by surgery on 2-knots, and to provide a reference for the topology of such manifolds and knots. The first chapter is purely algebraic. The rest of the book may be divided into three parts: general results on homotopy and surgery (Chapters 2-6), geometries and geometric decompositions (Chapters 7-13), and 2-knots (Chapters 14-18). In many cases the Euler characteristic, fundamental group and Stiefel-Whitney classes together form a complete system of invariants for the homotopy type of such manifolds, and the possible values of the invariants can be described explicitly. The strongest results are characterizations of manifolds which fibre homotopically over S^1 or an aspherical surface (up to homotopy equivalence) and infrasolvmanifolds (up to homeomorphism). As a consequence 2-knots whose groups are poly-Z are determined up to Gluck reconstruc...
Becker, Katrin; Robbins, Daniel
2015-01-01
In this talk we report on recent progress in describing compactifications of string theory and M-theory on G_2 and Spin(7) manifolds. We include the infinite set of alpha'-corrections and describe the entire tower of massless and massive Kaluza-Klein modes resulting from such compactifications.
Remarks on homogeneous manifolds satisfying Levi conditions
Huckleberry, Alan
2010-01-01
Homogeneous complex manifolds satisfying various types of Levi conditions are considered. Classical results which were of particular interest to Andreotti are recalled. Convexity and concavity properties of flag domains are discussed in some detail. A precise classification of pseudoconvex flag domains is given. It is shown that flag domains which are in a certain sense generic are pseudoconcave.
Exponential estimates of symplectic slow manifolds
DEFF Research Database (Denmark)
Kristiansen, Kristian Uldall; Wulff, C.
2016-01-01
is motivated by a paper of MacKay from 2004. The method does not notice resonances, and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic. We also present a stability result and obtain a generalization of a result of Gelfreich and Lerman...
Modelling of the Manifold Filling Dynamics
DEFF Research Database (Denmark)
Hendricks, Elbert; Chevalier, Alain Marie Roger; Jensen, Michael
1996-01-01
Mean Value Engine Models (MVEMs) are dynamic models which describe dynamic engine variable (or state) responses on time scales slightly longer than an engine event. This paper describes a new model of the intake manifold filling dynamics which is simple and easy to calibrate for use in engine con...
Heat Kernel Renormalization on Manifolds with Boundary
Albert, Benjamin I.
2016-01-01
In the monograph Renormalization and Effective Field Theory, Costello gave an inductive position space renormalization procedure for constructing an effective field theory that is based on heat kernel regularization of the propagator. In this paper, we extend Costello's renormalization procedure to a class of manifolds with boundary. In addition, we reorganize the presentation of the preexisting material, filling in details and strengthening the results.
Geometrical description of denormalized thermodynamic manifold
Institute of Scientific and Technical Information of China (English)
Wu Li-Ping; Sun Hua-Fei; Cao Li-Mei
2009-01-01
In view of differential geometry,the state space of thermodynamic parameters is investigated. Here the geometrical structures of the denormalized thermodynamic manifold are considered. The relation of their geometrical metrics is obtained. Moreover an example is used to illustrate our conclusions.
On homological stability for configuration spaces on closed background manifolds
Cantero, Federico; Palmer, Martin
2014-01-01
We introduce a new map between configuration spaces of points in a background manifold - the replication map - and prove that it is a homology isomorphism in a range with certain coefficients. This is particularly of interest when the background manifold is closed, in which case the classical stabilisation map does not exist. We then establish conditions on the manifold and on the coefficients under which homological stability holds for configuration spaces on closed manifolds. These conditio...
Royden's lemma in infinite dimensions and Hilbert-Hartogs manifolds
Ivashkovich, S
2011-01-01
We prove the Royden's Lemma for complex Hilbert manifolds, i.e., that a holomorphic imbedding of the closure of a finite dimensional, strictly pseudoconvex domain into a complex Hilbert manifold extends to a biholomorphic mapping onto a product of this domain with the unit ball in Hilbert space. This reduces several problems concerning complex Hilbert manifolds to open subsets of a Hilbert space. As an illustration we prove some results on generalized loop spaces of complex manifolds.
Fluid manifold design for a solar energy storage tank
Humphries, W. R.; Hewitt, H. C.; Griggs, E. I.
1975-01-01
A design technique for a fluid manifold for use in a solar energy storage tank is given. This analytical treatment generalizes the fluid equations pertinent to manifold design, giving manifold pressures, velocities, and orifice pressure differentials in terms of appropriate fluid and manifold geometry parameters. Experimental results used to corroborate analytical predictions are presented. These data indicate that variations in discharge coefficients due to variations in orifices can cause deviations between analytical predictions and actual performance values.
On Self-Mapping Degrees of S3- Geometry Manifolds
Institute of Scientific and Technical Information of China (English)
Xiao Ming DU
2009-01-01
In this paper we determined all of the possible self-mapping degrees of the manifolds with S3-geometry, which are supposed to be all 3-manifolds with finite fundamental groups. This is a part of a project to determine all possible self-mapping degrees of all closed orientable 3-manifold in Thurston's picture.
NUMERICAL MANIFOLD METHOD AND ITS APPLICATION IN UNDERGROUND POENINGS
Institute of Scientific and Technical Information of China (English)
王芝银; 李云鹏
1998-01-01
A brief introduction is made for the Numerical Manifold Method and its analysingprocess in rock mechanics. Some aspects of the manifold method are improved in implementingprocess according to the practice of excavating underground openings. Corresponding formulasare given and a computer program of the Numerical Manifold Method has been completed in thispaper.
Wave equations on anti self dual (ASD) manifolds
Bashingwa, Jean-Juste; Kara, A. H.
2017-06-01
In this paper, we study and perform analyses of the wave equation on some manifolds with non diagonal metric g_{ij} which are of neutral signatures. These include the invariance properties, variational symmetries and conservation laws. In the recent past, wave equations on the standard (space time) Lorentzian manifolds have been performed but not on the manifolds from metrics of neutral signatures.
Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey
Directory of Open Access Journals (Sweden)
Yvette Kosmann-Schwarzbach
2008-01-01
Full Text Available After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson-Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.
On the conformal geometry of transverse Riemann Lorentz manifolds
Aguirre, E.; Fernández, V.; Lafuente, J.
2007-06-01
Physical reasons suggested in [J.B. Hartle, S.W. Hawking, Wave function of the universe, Phys. Rev. D41 (1990) 1815-1834] for the Quantum Gravity Problem lead us to study type-changing metrics on a manifold. The most interesting cases are Transverse Riemann-Lorentz Manifolds. Here we study the conformal geometry of such manifolds.
Dynamics in Newtonian-Riemannian Space-Time(%Newton-Riemann时空中的动力学(Ⅳ)
Institute of Scientific and Technical Information of China (English)
张荣业
2001-01-01
Lagrangian mechanics in Newtonian-Riemannian space-time andrelationship between Lagrangian mechanics and Newtonian mechanics, and between Lagrangian mechanics and Hamiltonian mechanics in N-R space-time are discussed.%讨论了Newton-Riemann时空中的Lagrange力学及其与N-R时空中的Newton力学及Hamilton力学的关系.
Holst, Michael
2014-01-01
In this article we further develop the solution theory for the Einstein constraint equations on an n-dimensional, asymptotically Euclidean manifold M with interior boundary S. Building on recent results for both the asymptotically Euclidean and compact with boundary settings, we show existence of far-from-CMC and near-CMC solutions to the conformal formulation of the Einstein constraints when nonlinear Robin boundary conditions are imposed on S, similar to those analyzed previously by Dain (2004), by Maxwell (2004, 2005), and by Holst and Tsogtgerel (2013) as a model of black holes in various CMC settings, and by Holst, Meier, and Tsogtgerel (2013) in the setting of far-from-CMC solutions on compact manifolds with boundary. These "marginally trapped surface" Robin conditions ensure that the expansion scalars along null geodesics perpendicular to the boundary region S are non-positive, which is considered the correct mathematical model for black holes in the context of the Einstein constraint equations. Assumi...
On B-type open-closed Landau-Ginzburg theories defined on Calabi-Yau Stein manifolds
Babalic, Mirela; Lazaroiu, Calin Iuliu; Tavakol, Mehdi
2016-01-01
We consider the bulk algebra and topological D-brane category arising from the differential model of the open-closed B-type topological Landau-Ginzburg theory defined by a pair $(X,W)$, where $X$ is a non-compact Calabi-Yau manifold and $W$ has compact critical set. When $X$ is a Stein manifold (but not restricted to be a domain of holomorphy), we extract equivalent descriptions of the bulk algebra and of the category of topological D-branes which are constructed using only the analytic space associated to $X$. In particular, we show that the D-brane category is described by projective matrix factorizations defined over the ring of holomorphic functions of $X$. We also discuss simplifications of the analytic models which arise when $X$ is holomorphically parallelizable and illustrate these analytic models in a few classes of examples.
THE SOLUTION OF b的偏导-EQUATION OF （P,Q）-FORMS AND IT′ S Lp ＆ HOLDER ESTIMATES ON A STEIN MANIFOLD
Institute of Scientific and Technical Information of China (English)
WuXiaoqin
1994-01-01
By using the kernel of J. P. Demaily & Laurrent Thiebaut, the author constructs two opertors T and S which are compact and obtain the solution of b的偏导-equation of (p,q)forms and L & Holder estimates on a Stein Manifold.
Sparks, Rachel; Madabhushi, Anant
2012-03-01
Gleason patterns of prostate cancer histopathology, characterized primarily by morphological and architectural attributes of histological structures (glands and nuclei), have been found to be highly correlated with disease aggressiveness and patient outcome. Gleason patterns 4 and 5 are highly correlated with more aggressive disease and poorer patient outcome, while Gleason patterns 1-3 tend to reflect more favorable patient outcome. Because Gleason grading is done manually by a pathologist visually examining glass (or digital) slides, subtle morphologic and architectural differences of histological attributes may result in grading errors and hence cause high inter-observer variability. Recently some researchers have proposed computerized decision support systems to automatically grade Gleason patterns by using features pertaining to nuclear architecture, gland morphology, as well as tissue texture. Automated characterization of gland morphology has been shown to distinguish between intermediate Gleason patterns 3 and 4 with high accuracy. Manifold learning (ML) schemes attempt to generate a low dimensional manifold representation of a higher dimensional feature space while simultaneously preserving nonlinear relationships between object instances. Classification can then be performed in the low dimensional space with high accuracy. However ML is sensitive to the samples contained in the dataset; changes in the dataset may alter the manifold structure. In this paper we present a manifold regularization technique to constrain the low dimensional manifold to a specific range of possible manifold shapes, the range being determined via a statistical shape model of manifolds (SSMM). In this work we demonstrate applications of the SSMM in (1) identifying samples on the manifold which contain noise, defined as those samples which deviate from the SSMM, and (2) accurate out-of-sample extrapolation (OSE) of newly acquired samples onto a manifold constrained by the SSMM. We
Lower bounds on volumes of hyperbolic Haken 3-manifolds
Agol, Ian; Storm, Peter A.; Thurston, William P.
2007-10-01
We prove a volume inequality for 3-manifolds having C^{0} metrics ``bent'' along a surface and satisfying certain curvature conditions. The result makes use of Perelman's work on the Ricci flow and geometrization of closed 3-manifolds. Corollaries include a new proof of a conjecture of Bonahon about volumes of convex cores of Kleinian groups, improved volume estimates for certain Haken hyperbolic 3-manifolds, and a lower bound on the minimal volume of orientable hyperbolic 3-manifolds. An appendix compares estimates of volumes of hyperbolic 3-manifolds drilled along a closed embedded geodesic with experimental data.
THEORETICAL STUDY OF THREE-DIMENSIONAL NUMERICAL MANIFOLD METHOD
Institute of Scientific and Technical Information of China (English)
LUO Shao-ming; ZHANG Xiang-wei; L(U) Wen-ge; JIANG Dong-ru
2005-01-01
The three-dimensional numerical manifold method(NMM) is studied on the basis of two-dimensional numerical manifold method. The three-dimensional cover displacement function is studied. The mechanical analysis and Hammer integral method of three-dimensional numerical manifold method are put forward. The stiffness matrix of three-dimensional manifold element is derived and the dissection rules are given. The theoretical system and the numerical realizing method of three-dimensional numerical manifold method are systematically studied. As an example, the cantilever with load on the end is calculated, and the results show that the precision and efficiency are agreeable.
The cohomology of the affine Deligne-Lusztig varieties in the affine flag manifold of $GL_2$
Ivanov, Alexander
2009-01-01
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of GL_2. At first we determine all such varieties up to isomorphy. After this we investigate the representations of the sigma-stabilizer of an element b of the group on the etale cohomology of the affine Deligne-Lusztig variety X_w(b). We describe such representations as inductions from compact subgroups and in terms of noncuspidal representations.
Helical ${\\alpha}$-dynamos as twisted magnetic flux tubes in Riemannian space
de Andrade, Garcia
2007-01-01
Analytical solution of ${\\alpha}$-dynamo equation representing strongly torsioned helical dynamo is obtained in the thin twisted Riemannian flux tubes approximation. The $\\alpha$ factor possesses a fundamental contribution from torsion which is however weaken in the thin tubes approximation. It is shown that assuming that the poloidal component of the magnetic field is in principle time-independent, the toroidal magnetic field component grows very fast in time, actually it possesses a linear time dependence, while the poloidal component grows under the influence of torsion or twist of the flux tube. The toroidal component decays spatially with as $r^{-2}$ while vorticity may decay as $r^{-5}$ (poloidal component) where r represents the radial distance from the magnetic axis of flux tube. Toroidal component of vorticity decays as $r^{-1}$. In turbulent dynamos unbounded magnetic fields may decay at least as $r^{-3}$.
Turbulent ${\\alpha}$-effect in twisted magnetic flux tubes dynamos in Riemannian space
de Andrade, Garcia
2007-01-01
Analytical solution of first order torsion ${\\alpha}$-effect in twisted magnetic flux tubes representing a flux tube dynamo in Riemannian space is presented. Toroidal and poloidal component of the magnetic field decays as $r^{-1}$, while grow exponentially in time. The rate of speed of the helical dynamo depends upon the value of Frenet curvature of the tube. The $\\alpha$ factor possesses a fundamental contribution from constant torsion tube approximation. It is also assumed that the curvature of the magnetic axis of the tube is constant. Though ${\\alpha}$-effect dynamo equations are rather more complex in Riemann flux tube coordinates, a simple solution assuming force-free magnetic fields is shown to be possible. Dynamo solutions are possible if the dynamo action is able to change the signs of torsion and curvature of the dynamo flux tube simultaneously.
Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)
Valent, Galliano
2017-07-01
We present a family of superintegrable (SI) systems which live on a Riemannian surface of revolution and which exhibit one linear integral and two integrals of any integer degree larger or equal to 2 in the momenta. When this degree is 2, one recovers a metric due to Koenigs. The local structure of these systems is under control of a linear ordinary differential equation of order n which is homogeneous for even integrals and weakly inhomogeneous for odd integrals. The form of the integrals is explicitly given in the so-called "simple" case (see Definition 2). Some globally defined examples are worked out which live either in H2 or in R2.