On homogeneous submanifolds of negatively curved Riemannian manifolds

2013 ◽  
Vol 82 (2) ◽  
pp. 267-275 ◽  
Author(s):  
REZA MIRZAIE
Keyword(s):  
Riemannian Manifolds ◽  
Negatively Curved ◽  
Homogeneous Submanifolds

Some New Results on L2 Cohomology of Negatively Curved Riemannian Manifolds

2005 ◽  
Vol 57 (2) ◽  
pp. 251-266
Author(s):  
M. Cocos
Keyword(s):  
Heat Equation ◽  
Riemannian Manifolds ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Regularity Properties ◽  
L2 Cohomology ◽  
Cohomology Spaces

AbstractThe present paper is concerned with the study of the L2 cohomology spaces of negatively curved manifolds. The first half presents a finiteness and vanishing result obtained under some curvature assumptions, while the second half identifies a class of metrics having non-trivial L2 cohomology for degree equal to the half dimension of the space. For the second part we rely on the existence and regularity properties of the solution for the heat equation for forms.

scholarly journals Graph manifolds as ends of negatively curved Riemannian manifolds

2020 ◽  
Vol 24 (4) ◽  
pp. 2035-2074
Author(s):  
Koji Fujiwara ◽  
Takashi Shioya
Keyword(s):  
Riemannian Manifolds ◽  
Negatively Curved ◽  
Graph Manifolds

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 Lifting coarse homotopies

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Thomas Weighill
Keyword(s):  
Group Theory ◽  
Fundamental Group ◽  
Riemannian Manifolds ◽  
Homotopy Theory ◽  
Index Theory ◽  
Geometric Group Theory ◽  
Coarse Geometry ◽  
Discontinuous Group ◽  
Negatively Curved ◽  
Develop Theory

Abstract Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper,we continue to develop theory in this area by proving a Coarse Lifting Lemma with respect to a certain class of bornologous surjective maps. This class is wide enough to include quotients by coarsely discontinuous group actions, which allows us to obtain results concerning the coarse fundamental group of quotients which are analogous to classical topological results for the fundamental group. As an application, we compute the fundamental group of metric cones over negatively curved compact Riemannian manifolds.

scholarly journals Entropy, minimal surfaces and negatively curved manifolds

2016 ◽  
Vol 38 (1) ◽  
pp. 336-370
Author(s):  
ANDREW SANDERS
Keyword(s):  
Hausdorff Dimension ◽  
Riemannian Manifolds ◽  
Minimal Surfaces ◽  
Second Fundamental Form ◽  
Fuchsian Groups ◽  
Riemannian Surface ◽  
Limit Set ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Convex Cocompact

Taubes [Minimal surfaces in germs of hyperbolic 3-manifolds. Proceedings of the Casson Fest, Geom. Topol. Monogr.7 (2004), 69–100 (electronic)] introduced the space of minimal hyperbolic germs with elements consisting of the first and second fundamental form of an equivariant immersed minimal disk in hyperbolic 3-space. Herein, we initiate a further study of this space by studying the behavior of a dynamically defined function which records the entropy of the geodesic flow on the associated Riemannian surface. We provide a useful estimate on this function which, in particular, yields a new proof of Bowen’s theorem on the rigidity of the Hausdorff dimension of the limit set of quasi-Fuchsian groups. These follow from new lower bounds on the Hausdorff dimension of the limit set which allow us to give a quantitative version of Bowen’s rigidity theorem. To demonstrate the strength of the techniques, these results are generalized to convex-cocompact surface groups acting on $n$-dimensional $\text{CAT}\,(-1)$ Riemannian manifolds.

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