Ergodicity of Geodesic Flows

Consider a point moving on a frictionless curved surface after being given an initial position and velocity. Assuming that there are no external forces, the point will follow a unique "straight line" path on the surface. This fact generates an important dynamical system known as the geodesic flow. How does the curvature of the surface affect the behavior of this flow? After giving some ergodic theory background, I will present E. Hopf's seminal theorem that the geodesic flow of a compact surface of constant negative curvature is "globally complicated" in the sense that it is ergodic. I will conclude by mentioning some generalizations and open questions.