Some Rigidity Theorem for Anosov 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.