On the role of the surface geometry in convex billiards

Abstract This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some properties that have long been known for billiards on the plane are established. We prove the twist property of the billiard maps and establish some conditions for the existence and non-existence of rotational invariant curves. Although we prove that Lazutkin’s and Hubacher’s theorems are valid for general surfaces, we also find that Mather’s theorem does not apply to surfaces of non-negative curvature.