Scattering rigidity with trapped geodesics

TL;DR

This paper proves that the flat product metric on the product of a unit ball and a circle is uniquely determined by its scattering data, even with trapped geodesics, establishing a new rigidity result in Riemannian geometry.

Contribution

It demonstrates scattering rigidity for a manifold with trapped geodesics, a first in the field, by showing boundary data determines the interior metric.

Findings

The flat product metric on $D^n\times S^1$ is scattering rigid.

Manifolds with the same scattering data and boundary areometric to $D^n\times S^1$ are isometric.

Trapped geodesics have measure zero in the unit tangent bundle.

Abstract

We prove that the flat product metric on $D^n\times S^1$ is scattering rigid where $D^n$ is the unit ball in $\R^n$ and $n\geq 2$. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map $S:U^+\partial M\to U^-\partial M$ from unit vectors $V$ at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes $V$ to $\gamma'_V(T_0)$ where $\gamma_V$ is the unit speed geodesic determined by $V$ and $T_0$ is the first positive value of $t$ (when it exists) such that $\gamma_V(t)$ again lies in the boundary. We show that any other Riemannian manifold $(M,\partial M,g)$ with boundary $\partial M$ isometric to $\partial(D^n\times S^1)$ and with the same scattering data must be isometric to $D^n\times S^1$. This is the first scattering rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in $(M,\partial M,g)$ have measure 0 in the unit tangent bundle.