Another Billiard Problem

Let \((M,g)\) be a Riemannian manifold, \(\Omega\subset M\) a domain with boundary \(\Gamma\), and \(\phi\) a smooth function such that \(\phi|_\Omega > 0\), \( \varphi |_\Gamma = 0\), and \(d\phi|_\Gamma\ne 0\). 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 (not quite standard) billiard problem in \(\Omega\).

DOI 10.1134/S106192084010047