On the role of the surface geometry in convex billiards (Q6623097)

The authors consider billiards on two-dimensional Riemann manifolds, such that their domains are contained in simply connected open sets which are totally normal. Some well-known properties of planar billiards are investigated in this general setting. The twist property of the billiard map is proved and some conditions for existence and non-existence of rational invariant curves established. Generalisations of Lazutkin's and Hubacher's theorems are obtained. Mather's theorem is proved on surfaces with non-positive Gaussian curvature and for sufficiently small billiard domains otherwise. On a surface with positive Gaussian curvature, an example of a billiard table with a point of zero geodesic curvature on the boundary, such that the billiard map has an invariant rotational curve is constructed.