scholarly journals Simplicial volume of compact manifolds with amenable boundary

2014 ◽  
Vol 07 (01) ◽  
pp. 23-46 ◽  
Author(s):  
Sungwoon Kim ◽  
Thilo Kuessner
Keyword(s):  
Fundamental Group ◽  
Finite Volume ◽  
Compact Manifold ◽  
Riemannian Metric ◽  
Compact Manifolds ◽  
Simplicial Volume ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Path Component ◽  
Proportionality Principle

Let M be the interior of a connected, oriented, compact manifold V of dimension at least 2. If each path component of ∂V has amenable fundamental group, then we prove that the simplicial volume of M is equal to the relative simplicial volume of V and also to the geometric (Lipschitz) simplicial volume of any Riemannian metric on M whenever the latter is finite. As an application we establish the proportionality principle for the simplicial volume of complete, pinched negatively curved manifolds of finite volume.

Regularity at infinity of compact negatively curved manifolds

1994 ◽  
Vol 14 (3) ◽  
pp. 493-514
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Tangent Bundle ◽  
Compact Manifold ◽  
Negative Curvature ◽  
Periodic Points ◽  
Geodesic Flows ◽  
Stable Foliation ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Unit Tangent Bundle ◽  
Measure Class

AbstractIt is shown that three different notions of regularity for the stable foliation on the unit tangent bundle of a compact manifold of negative curvature are equivalent. Moreover if is a time-preserving conjugacy of geodesic flows of such manifolds M, N then the Lyapunov exponents at corresponding periodic points of the flows coincide. In particular Δ also preserves the Lebesgue measure class.

scholarly journals Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres

2019 ◽  
Vol 41 (2) ◽  
pp. 553-569 ◽  
Author(s):  
CHRISTOFOROS NEOFYTIDIS
Keyword(s):  
Homology Sphere ◽  
Homotopy Groups ◽  
Rational Homology ◽  
Simplicial Volume ◽  
Anosov Diffeomorphisms ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Rational Homology Spheres ◽  
Aspherical Manifolds ◽  
Rational Homology Sphere

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle–Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimension at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, including certain manifolds with non-trivial higher homotopy groups and certain products of aspherical manifolds.

scholarly journals Asymptotic geometry of negatively curved manifolds of finite volume

2019 ◽  
Vol 52 (6) ◽  
pp. 1459-1485
Author(s):  
Françoise DAL'BO ◽  
Marc PEIGNE ◽  
Jean-Claude PICAUD ◽  
Andrea SAMBUSETTI
Keyword(s):  
Finite Volume ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Asymptotic Geometry

scholarly journals Entropy rigidity of negatively curved manifolds of finite volume

2018 ◽  
Vol 293 (1-2) ◽  
pp. 609-627 ◽  
Author(s):  
M. Peigné ◽  
A. Sambusetti
Keyword(s):  
Finite Volume ◽  
Curved Manifolds ◽  
Negatively Curved

scholarly journals Transformation Groups with (n-1)-Dimensional Orbits on Non-Compact Manifolds

1959 ◽  
Vol 14 ◽  
pp. 25-38 ◽  
Author(s):  
Tadashi Nagano
Keyword(s):  
Lie Group ◽  
Compact Manifold ◽  
Isometry Group ◽  
Transformation Group ◽  
Differentiable Manifold ◽  
Riemannian Metric ◽  
Compact Manifolds ◽  
Transformation Groups ◽  
Lie Transformation

When a Lie group G operates on a differentiable manifold M as a Lie transformation group, the orbit of a point p in M under G, or the G-orbit of p, is by definition the submanifold G(p) = {G(p); g∈G}. The purpose of this paper is to characterize the structure of a non-compact manifold M such that there exists a compact orbit of dimension (n — 1), n — dim M, under a connected Lie transformation group G, which is assumed to be compact or an isometry group of a Riemannian metric on M.

scholarly journals Piecewise straightening and Lipschitz simplicial volume

2017 ◽  
Vol 09 (01) ◽  
pp. 167-193
Author(s):  
Karol Strzałkowski
Keyword(s):  
Sectional Curvature ◽  
Finite Volume ◽  
Riemannian Manifolds ◽  
Simplicial Volume ◽  
Product Inequality ◽  
Complete Riemannian Manifolds ◽  
Proportionality Principle

We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with sectional curvature bounded from above.

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