Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold

2001 ◽  
Vol 77 (6) ◽  
pp. 522-528 ◽  
Author(s):  
J. Latschev
Keyword(s):  
Riemannian Manifold ◽  
Metric Spaces ◽  
Closed Riemannian Manifold

On the Inner Radius of a Nodal Domain

2008 ◽  
Vol 51 (2) ◽  
pp. 249-260 ◽  
Author(s):  
Dan Mangoubi
Keyword(s):  
Riemannian Manifold ◽  
Lower Bounds ◽  
Type Inequality ◽  
Large Eigenvalue ◽  
Upper And Lower Bounds ◽  
Local Behavior ◽  
Nodal Domain ◽  
Closed Riemannian Manifold ◽  
Inner Radius

AbstractLet M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue λ. We give upper and lower bounds on the inner radius of the type C/λα(log λ)β. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincaré type inequality proved by Maz’ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.

scholarly journals Focal rigidity of flat tori

2011 ◽  
Vol 83 (4) ◽  
pp. 1149-1158 ◽  
Author(s):  
Ferry Kwakkel ◽  
Marco Martens ◽  
Mauricio Peixoto
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Topological Structure ◽  
Closed Riemannian Manifold ◽  
Flat Tori

Given a closed Riemannian manifold (M, g), i.e. compact and boundaryless, there is a partition of its tangent bundle TM = ∪iΣi called the focal decomposition of TM. The sets Σi are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that flat n-tori, n > 2, are focally rigid in the sense that if two flat tori are focally equivalent then the tori are isometric up to rescaling. The case n = 2 was considered before by F. Kwakkel.

A coarse Mayer–Vietoris principle

1993 ◽  
Vol 114 (1) ◽  
pp. 85-97 ◽  
Author(s):  
Nigel Higson ◽  
John Roe ◽  
Guoliang Yu
Keyword(s):  
Riemannian Manifold ◽  
Metric Space ◽  
Metric Spaces ◽  
Compact Operators ◽  
Cohomology Theory ◽  
Coarse Geometry ◽  
Locally Compact ◽  
Finite Propagation ◽  
Noncompact Riemannian Manifold

In [1], [4], and [6] the authors have studied index problems associated with the ‘coarse geometry’ of a metric space, which typically might be a complete noncompact Riemannian manifold or a group equipped with a word metric. The second author has introduced a cohomology theory, coarse cohomology, which is functorial on the category of metric spaces and coarse maps, and which can be computed in many examples. Associated to such a metric space there is also a C*-algebra generated by locally compact operators with finite propagation. In this note we will show that for suitable decompositions of a metric space there are Mayer–Vietoris sequences both in coarse cohomology and in the K-theory of the C*-algebra. As an application we shall calculate the K-theory of the C*-algebra associated to a metric cone. The result is consistent with the calculation of the coarse cohomology of the cone, and with a ‘coarse’ version of the Baum–Connes conjecture.

scholarly journals Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations

1988 ◽  
Vol 8 (2) ◽  
pp. 215-239 ◽  
Author(s):  
Masahiko Kanai
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

AbstractWe are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

scholarly journals Metric completions, the Heine-Borel property, and approachability

2020 ◽  
Vol 18 (1) ◽  
pp. 162-166
Author(s):  
Vladimir Kanovei ◽  
Mikhail G. Katz ◽  
Tahl Nowik
Keyword(s):  
Riemannian Manifold ◽  
Metric Space ◽  
Metric Spaces ◽  
Short Proof ◽  
Universal Cover

Abstract We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by Do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in ∗M.

scholarly journals Semiclassical states for a static supercritical Klein–Gordon–Maxwell–Proca system on a closed Riemannian manifold

2016 ◽  
Vol 18 (03) ◽  
pp. 1550039 ◽  
Author(s):  
Mónica Clapp ◽  
Marco Ghimenti ◽  
Anna Maria Micheletti
Keyword(s):  
Riemannian Manifold ◽  
Mass Formula ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Perturbation Parameter ◽  
Positive Dimension ◽  
Closed Riemannian Manifold ◽  
Klein Gordon ◽  
Semiclassical States ◽  
Static Form

We establish the existence of semiclassical states for a nonlinear Klein–Gordon–Maxwell–Proca system in static form, with Proca mass [Formula: see text] on a closed Riemannian manifold. Our results include manifolds of arbitrary dimension and allow supercritical nonlinearities. In particular, we exhibit a large class of three-dimensional manifolds on which the system has semiclassical solutions for every exponent [Formula: see text] The solutions we obtain concentrate at closed submanifolds of positive dimension as the singular perturbation parameter goes to zero.

scholarly journals Volumes of balls in Riemannian manifolds and Uryson width

2017 ◽  
Vol 09 (02) ◽  
pp. 195-219 ◽  
Author(s):  
Larry Guth
Keyword(s):  
Riemannian Manifold ◽  
Unit Ball ◽  
Riemannian Manifolds ◽  
Closed Riemannian Manifold ◽  
Small Constant

If [Formula: see text] is a closed Riemannian manifold where every unit ball has volume at most [Formula: see text] (a sufficiently small constant), then the [Formula: see text]-dimensional Uryson width of [Formula: see text] is at most 1.

scholarly journals Convergence of the solutions of discounted Hamilton–Jacobi systems

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrea Davini ◽  
Maxime Zavidovique
Keyword(s):  
Riemannian Manifold ◽  
Coupled System ◽  
Discount Factor ◽  
Limit System ◽  
Specific Solution ◽  
Closed Riemannian Manifold ◽  
Weakly Coupled ◽  
Weakly Coupled System ◽  
Jacobi Equations ◽  
Minimizing Measures

Abstract We consider a weakly coupled system of discounted Hamilton–Jacobi equations set on a closed Riemannian manifold. We prove that the corresponding solutions converge to a specific solution of the limit system as the discount factor goes to 0. The analysis is based on a generalization of the theory of Mather minimizing measures for Hamilton–Jacobi systems and on suitable random representation formulae for the discounted solutions.

scholarly journals ENERGY OF GLOBAL FRAMES

2008 ◽  
Vol 84 (2) ◽  
pp. 155-162
Author(s):  
FABIANO G. B. BRITO ◽  
PABLO M. CHACÓN
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Lower Bound ◽  
Vector Fields ◽  
Singular Vector ◽  
Unit Vector ◽  
Energy Functional ◽  
Closed Riemannian Manifold ◽  
Total Scalar Curvature ◽  
Dimension 2

AbstractThe energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k+1 which is not attained by any non-singular vector field for k>1. For k=1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

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