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 is of Anosov type, then the constant of contraction of the flow is greater than 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 for bi-Holder, bi-Lipschitz, and consequently, for C1-conjugacy between two geodesic flows.