Another billiard problem

Abstract

Let $(M,g)$ be a Riemannian manifold, $\hat{I}\text{c◯}\subset M$ a domain with boundary $\hat{I}“$, and $\ddot{I}•$ a smooth function such that $\ddot{I}•{|}_{\hat{I}}\text{c◯}>0$, $\ph|_Γ= 0$, and $\nabla \ddot{I}•{|}_{\hat{I}}“\ne 0$. We study the geodesic flow of the metric $G=g/\ddot{I}•$. The $G$-distance from any point of $\hat{I}\text{c◯}$ to $\hat{I}“$ is finite, hence the geodesic flow is incomplete. Regularization of the flow in a neighborhood of $\hat{I}“$ establishes a natural reflection law from $\hat{I}“$. This leads to a certain billiard problem in $\hat{I}\text{c◯}$.