It was proved by Donnay that any compact orientable surface can be given a Riemannian metric for which the geodesic flow is ergodic.See Theorem 1 of this article.

It was proved by Donnay that any compact orientable surface can be given a Riemannian metric for which the geodesic flow is ergodic. See Theorem 1 of this article. On the other hand, there are no negatively curved metrics on a sphere or a torus.

It was proved by Donnay that any compact orientable surface can be given a Riemannian metric for which the geodesic flow is ergodic.

It was proved by Donnay that any compact orientable surface can be given a Riemannian metric for which the geodesic flow is ergodic.See Theorem 1 of this article.