Some Rigidity Theorem for Anosov Geodesic Flows

TL;DR

This paper establishes a rigidity theorem for Anosov geodesic flows on manifolds without conjugate points, linking contraction constants to curvature bounds and characterizing when the curvature is constant.

Contribution

It proves a new rigidity result relating contraction constants of Anosov geodesic flows to sectional curvature bounds and characterizes cases of equality for finite volume manifolds.

Findings

Contraction constant of geodesic flow is at least e^{-c} for manifolds with curvature ≥ -c^2.

Equality of contraction constant and e^{-c} characterizes constant sectional curvature in finite volume manifolds.

Results imply rigidity of bi-Lipschitz and C^1-conjugacies between geodesic flows.

Abstract

In this paper, we prove that if the geodesic flow of a complete manifold without conjugate points with sectional curvatures bounded below by $-c^2$ is of Anosov type, then the constant of contraction of the flow is $\geq e^{-c}$. Moreover, if $M$ has finite volume, the equality holds if and only if the sectional curvature is constant. We also apply this result to get a certain rigidity bi-Lipschitz conjugation, and consequently, for $C^1$-conjugacy between two geodesic flows.