Another Billiard Problem
Sergey Vladimirovich Bolotin, Dmitry Valerevich Treschev · Russian Journal of Mathematical Physics · 2024
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 $$d\phi|_\Gamma e 0$$ . 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