Another billiard problem

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