  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.
