Rigidity of Travelling Times for Strictly Convex Obstacles in Riemannian Manifolds

Abstract

Let $K$ and $L$ be two disjoint unions of strictly convex obstacles contained within a Riemannian manifold with boundary $S$ of dimension $m\ge 2$. The sets of travelling times ${\mathcal{T}}_K$ and ${\mathcal{T}}_L$ of $K$ and $L$, respectively, are composed of triples $(x,y,t)\in \partial S\times \partial S\times {\mathbb{R}}^{+}$ where $t$ is the length of a billiard trajectory with endpoints $x$ and $y$ that reflects elastically on $K$ (or $L$ for $(x,y,t)\in {\mathcal{T}}_L$). It has been shown (arXiv:2309.11141) that (under some natural curvature bounds on $S$) if ${\mathcal{T}}_K={\mathcal{T}}_L$ and $K$ and $L$ were equivalent up to tangency then $K=L$. In this paper we remove this requirement for $K$ and $L$, and show that if ${\mathcal{T}}_K={\mathcal{T}}_L$ then $K=L$ whenever $m\ge 3$.