  In this paper we show that the geodesic flow of a Finsler metric is Anosov if
and only if there exists a $C^2$ open neighborhood of Finsler metrics all of
whose closed geodesics are hyperbolic. For surfaces this result holds also for
Riemannian metrics. This follows from a recent result of Contreras and
Mazzucchelli. Furthermore, geodesic flows of Riemannian or Finsler metrics on
surfaces are $C^2$ stably ergodic if and only if they are Anosov.
