Geodesic Flows on Manifolds of Negative Curvature with Smooth Horospheric Foliations

Citation

Feres, Renato (1989) Geodesic Flows on Manifolds of Negative Curvature with Smooth Horospheric Foliations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/f6yt-bf73. https://resolver.caltech.edu/CaltechETD:etd-05232007-115904

Abstract

We improve a result due to M. Kanai on the rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable (horospheric) foliation is smooth. More precisely, the main result proven here is: Let M be a closed C^∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φ_t: 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.

Thesis Files