Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows

We consider in this note smooth dynamical systems equipped with smooth invariant affine connections and show that, under a pinching condition on the Lyapunov exponents, certain invariant tensor fields are parallel. We then apply this result to a problem of rigidity of geodesic flows for Riemannian manifolds with negative curvature.