  We consider a billiard in the sphere S^2 with circular obstacles, and give a
sufficient condition for its flow to be uniformly hyperbolic. We show that the
billiard flow in this case is approximated by an Anosov geodesic flow on a
surface in the ambiant space S^3. As an application, we show that every
orientable surface of genus at least 11 admits an isometric embedding into S^3
(equipped with the standard metric) such that its geodesic flow is Anosov.
Finally, we explain why this construction cannot provide examples of isometric
embeddings of surfaces in the Euclidean R^3 with Anosov geodesic flows.

Dynamical Systems;Differential Geometry;Geometric Topology