Some Rigidity Theorems for Anosov Geodesic Flows in Manifolds of Finite Volume

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