Another billiard problem

TL;DR

This paper investigates a billiard problem derived from the geodesic flow on a Riemannian manifold with a modified metric, establishing a reflection law at the boundary through flow regularization.

Contribution

It introduces a novel billiard problem based on geodesic flow with a conformally changed metric and develops a regularization method to define boundary reflections.

Findings

Finite G-distance to boundary leads to incomplete flow.

Regularization yields a natural reflection law.

Formulation of a new billiard problem in Riemannian geometry.

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 $\Omega$.