Geodesic Flows on Manifolds of Negative Curvature with Smooth Horospheric Foliations

Author: Feres, Renato

Year: 1989

Degree: Dissertation (Ph.D.)

Advisor: Kechris, Alexander S.

Committee Members: Katok, Anatole; Luxemburg, W. A. J.; Ramakrishnan, Dinakar; Kechris, Alexander S.

Option: Mathematics

DOI: 10.7907/f6yt-bf73

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 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.

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