Geodesic flows on manifolds of constant negative curvature

1955 ◽  
pp. 49-65 ◽  
Author(s):  
I. M. Gel′fand ◽  
S. V. Fomin
Keyword(s):  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature

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.

ELLIPTIC OPERATORS AND SOLUTIONS OF COHOMOLOGICAL EQUATIONS FOR GEODESIC FLOWS WITH HYPERBOLIC BEHAVIOR

1991 ◽  
Vol 02 (06) ◽  
pp. 701-709
Author(s):  
SVETLANA KATOK
Keyword(s):  
Negative Curvature ◽  
Infinitesimal Generator ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Smooth Functions ◽  
Negatively Curved ◽  
Finite Dimensional ◽  
Unit Tangent Bundle ◽  
Hyperbolic Behavior ◽  
Cohomological Equations

In this paper we study the space of smooth functions on the unit tangent bundle SM to a compact negatively curved surface M that are eigenfunctions of the infinitesimal generator of the action of SO(2) on SM, and that have zero integrals over all periodic orbits of the geodesic flow on SM. It is proved that the space of such functions is finite dimensional. In the case of constant negative curvature a complete description of this space is obtained.

scholarly journals Geodesic flows on manifolds of negative curvature with smooth horospheric foliations

1991 ◽  
Vol 11 (4) ◽  
pp. 653-686 ◽  
Author(s):  
Renato Feres
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Negative Sectional Curvature

AbstractWe improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φt: V → V on the unit tangent bundle V of M is C∞. Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C∞-isomorphic (i.e., conjugate under a C∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ1, is C∞-isomorphic to the flow , on the quotient Γ\, where Γ is a subgroup of a real Lie group ⊂ Diffeo () with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂHm, m = ½ dim M.

scholarly journals Nondense orbits of flows on homogeneous spaces

1998 ◽  
Vol 18 (2) ◽  
pp. 373-396 ◽  
Author(s):  
DMITRY Y. KLEINBOCK
Keyword(s):  
Hausdorff Dimension ◽  
Homogeneous Space ◽  
Lie Group ◽  
Negative Curvature ◽  
Cyclic Subgroup ◽  
Homogeneous Spaces ◽  
Discrete Subgroup ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Set Of Points

Let $F$ be a nonquasi-unipotent one-parameter (cyclic) subgroup of a unimodular Lie group $G$, $\Gamma$ a discrete subgroup of $G$. We prove that for certain classes of subsets $Z$ of the homogeneous space $G/\Gamma$, the set of points in $G/\Gamma$ with $F$-orbits staying away from $Z$ has full Hausdorff dimension. From this we derive applications to geodesic flows on manifolds of constant negative curvature.

scholarly journals Flots d'Anosov sur les 3-variétés fibrées en cercles

1984 ◽  
Vol 4 (1) ◽  
pp. 67-80 ◽  
Author(s):  
Etienne Ghys
Keyword(s):  
Geodesic Flow ◽  
Negative Curvature ◽  
Hyperbolic Manifold ◽  
Geodesic Flows ◽  
Finite Covering ◽  
Constant Negative Curvature ◽  
Circle Bundles ◽  
Anosov Flows

AbstractWe consider Anosov flows on closed 3-manifolds which are circle bundles. Our main result is that, up to a finite covering, these flows are topologically equivalent to the geodesic flow of a suface of constant negative curvature. The same method shows that, if M is a closed hyperbolic manifold of any dimension, all the geodesic flows which correspond to different metrics on M and which are of Anosov type are topologically equivalent.

scholarly journals Rational ergodicity of geodesic flows

1984 ◽  
Vol 4 (2) ◽  
pp. 165-178 ◽  
Author(s):  
Jon Aaronson ◽  
Dennis Sullivan
Keyword(s):  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Divergence Type

AbstractWe prove the rational egodicity of geodesic flows on divergence type surfaces of constant negative curvature, and identify their asymptotic types.

Sign in / Sign up

Export Citation Format

Share Document