  Uniform hyperbolicity is a strong chaotic property which holds, in
particular, for Sinai billiards. In this paper, we consider the case of a
nonflat billiard, that is, a Riemannian manifold with boundary. Each trajectory
follows the geodesic flow in the interior of the billiard, and bounces when it
meets the boundary. We give a sufficient condition for a nonflat billiard to be
uniformly hyperbolic. As a particular case, we obtain a new criterion to show
that a closed surface has an Anosov geodesic flow.
