Another billiard problem

Abstract

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