Uniqueness of Obstacles in Riemannian Manifolds from Travelling Times

TL;DR

This paper proves that, under certain curvature and intersection conditions, two disjoint unions of convex obstacles in a Riemannian manifold with identical travelling times must be the same set.

Contribution

It establishes the uniqueness of obstacle configurations in Riemannian manifolds based on travelling time data under specific geometric constraints.

Findings

Obstacles with identical travelling times are identical under curvature conditions.

Uniqueness holds when no geodesic intersects more than two obstacle components.

Results extend previous work to more general Riemannian settings.

Abstract

Suppose that $K$ and $L$ are two disjoint unions of strictly convex obstacles with the same set of travelling times, contained in an $n$-dimensional Riemannian manifold $M$ (where $n\geq2$). Under some natural curvature conditions on $M$, and provided that no geodesic intersects more than two components in $K$ or $L$, we show that $K = L$.