Topological stability and Gromov hyperbolicity

We show that if the geodesic flow of a compact analytic Riemannian manifold M of non-positive curvature is either C-k-topologically stable or satisfie s the epsilon-C-k-shadowing property for some k > 0 then the universal cove ring of M is a Gromov hyperbolic space. The same holds for compact surfaces without conjugate points.