scholarly journals Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary II

2020 ◽  
Vol 28 (1) ◽  
pp. 165-179
Author(s):  
Luca Sabatini
Keyword(s):  
Riemannian Manifold ◽  
Lower Bound ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Injectivity Radius ◽  
Topological Invariants ◽  
Manifolds With Boundary ◽  
Laplacian Spectrum ◽  
Convex Boundary

AbstractWe present some estimate of the Laplacian Spectrum and of Topological Invariants for Riemannian manifold with pinched sectional curvature and with non-empty and non-convex boundary with finite injectivity radius. These estimates do not depend directly on the the lower bound of the boundary injectivity radius but on the bounds of the curvatures of the manifold and its boundary.

scholarly journals Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary

2019 ◽  
Vol 27 (2) ◽  
pp. 179-211
Author(s):  
Luca Sabatini
Keyword(s):  
Riemannian Manifold ◽  
Lower Bound ◽  
Riemannian Manifolds ◽  
Topological Invariants ◽  
Manifolds With Boundary ◽  
Laplacian Spectrum ◽  
Common Boundary

AbstractWe set out to obtain estimates of the Laplacian Spectrum of Riemannian manifolds with non-empty boundary. This was achieved using standard doubled manifold techniques. In simple terms, we pasted two copies of the same manifold along their common boundary thereby obtaining a Riemannian manifold with empty boundary and with a C0−metric. This made it possible to adapt some estimates of the spectrum dependent on the volume or genus of the manifold as calculated in recent years by several authors. In order to extend further estimates that depend on the curvature, it is necessary to regularize the metric of the doubled manifold so that the new metric is isometric to that of each copy and such that the curvature has a finite lower bound. Controlling the curvature in this way also makes estimates of topological invariants available.

Closed geodesics in Riemannian manifolds with convex boundary

1994 ◽  
Vol 124 (6) ◽  
pp. 1247-1258 ◽  
Author(s):  
Anna Maria Candela ◽  
Addolorata Salvatore
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Closed Geodesics ◽  
Convex Boundary

In this paper we look for closed geodesies on a noncomplete Riemannian manifold ℳ. We prove that if ℳ has convex boundary, then there exists at least one closed nonconstant geodesic on it.

scholarly journals On the number of diffeomorphism classes in a certain class of Riemannian manifolds

1985 ◽  
Vol 97 ◽  
pp. 173-192 ◽  
Author(s):  
Takao Yamaguchi
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Homotopy Types

The study of finiteness for Riemannian manifolds, which has been done originally by J. Cheeger [5] and A. Weinstein [13], is to investigate what bounds on the sizes of geometrical quantities imply finiteness of topological types, —e.g. homotopy types, homeomorphism or diffeomorphism classes-— of manifolds admitting metrics which satisfy the bounds. For a Riemannian manifold M we denote by RM and KM respectively the curvature tensor and the sectional curvature, by Vol (M) the volume, and by diam(M) the diameter.

Curvature of K-contact Semi-Riemannian Manifolds

2014 ◽  
Vol 57 (2) ◽  
pp. 401-412 ◽  
Author(s):  
Domenico Perrone
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Lorentzian Manifold ◽  
Constant Sectional Curvature ◽  
Conformally Flat ◽  
Reeb Vector ◽  
Reeb Vector Field

Abstract.In this paper we characterize K-contact semi-Riemannian manifolds and Sasakian semi- Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature κ = ɛ, where ɛ = ± denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a K-contact Lorentzian manifold.

Existence of a Closed Geodesic on Non-compact Riemannian Manifolds with Boundary

2002 ◽  
Vol 2 (1) ◽  
Author(s):  
Rossella Bartolo ◽  
Anna Germinario ◽  
Miguel Sánchez
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Closed Geodesic ◽  
Detailed Comparison ◽  
Manifolds With Boundary ◽  
Manifold With Boundary

AbstractA new result about the existence of a closed geodesic on a Riemannian manifold with boundary is given. A detailed comparison with previous results is carried out.

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 A Second Order Smooth Variational Principle on Riemannian Manifolds

2010 ◽  
Vol 62 (2) ◽  
pp. 241-260 ◽  
Author(s):  
Daniel Azagra ◽  
Robb Fry
Keyword(s):  
Variational Principle ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Second Order ◽  
Injectivity Radius ◽  
Infinite Dimensional ◽  
Locally Convex

AbstractWe establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

EXTREMAL FUNCTIONS FOR MOSER–TRUDINGER INEQUALITIES ON 2-DIMENSIONAL COMPACT RIEMANNIAN MANIFOLDS WITH BOUNDARY

2006 ◽  
Vol 17 (03) ◽  
pp. 313-330 ◽  
Author(s):  
YUNYAN YANG
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Extremal Functions ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Blowing Up

Let (M,g) be a 2-dimensional compact Riemannian manifold with boundary. In this paper, we use the method of blowing up analysis to prove the existence of the extremal functions for some Moser–Trudinger inequalities on (M,g).

scholarly journals Reconstruction and Interpolation of Manifolds. I: The Geometric Whitney Problem

2019 ◽  
Vol 20 (5) ◽  
pp. 1035-1133
Author(s):  
Charles Fefferman ◽  
Sergei Ivanov ◽  
Yaroslav Kurylev ◽  
Matti Lassas ◽  
Hariharan Narayanan
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Metric Space ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Injectivity Radius ◽  
Principal Curvatures ◽  
Curvature Bounds ◽  
Fixed Dimension ◽  
Manifold Reconstruction

Abstract We study the geometric Whitney problem on how a Riemannian manifold (M, g) can be constructed to approximate a metric space $$(X,d_X)$$ ( X , d X ) . This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold $$S\subset {{\mathbb {R}}}^m$$ S ⊂ R m , $$m>n$$ m > n needs to be constructed to approximate a point cloud in $${{\mathbb {R}}}^m$$ R m . These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov–Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results, we give a new characterization of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterize the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalization of the Whitney embedding construction where approximative coordinate charts are embedded in $${{\mathbb {R}}}^m$$ R m and interpolated to a smooth submanifold.

Prescribed k-Curvature Problems on Complete Noncompact Riemannian Manifolds

2018 ◽  
Vol 2020 (23) ◽  
pp. 9559-9592
Author(s):  
Jixiang Fu ◽  
Weimin Sheng ◽  
Lixia Yuan
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Elliptic Partial Differential Equations ◽  
Nonlinear Elliptic ◽  
Noncompact Riemannian Manifolds ◽  
Curvature Tensors ◽  
Curvature Problem ◽  
Negative Sectional Curvature

Abstract To study the prescribed $k$-curvature problem of 2nd-order symmetric curvature tensors on complete noncompact Riemannian manifolds, we consider a class of fully nonlinear elliptic partial differential equations. It is proved that on a Riemannian manifold with negative sectional curvature and Ricci curvature bounded from below, the equation is solvable provided that all the eigenvalues of the tensor are negative. The result is applicable to the prescribed $k$-curvature problems of modified Schouten tensor and Bakry–Émery Ricci tensor.

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