Local lens rigidity for manifolds of Anosov type

TL;DR

This paper proves local lens rigidity for negatively curved Riemannian manifolds with convex boundary, showing that the lens data uniquely determines the metric up to isometry within certain classes.

Contribution

It extends local lens rigidity results to manifolds of Anosov type, including higher dimensions and non-positive curvature cases.

Findings

Local lens data determines the metric up to isometry for negatively curved manifolds.

Rigidity holds for Anosov type metrics in dimension 2 and higher with non-positive curvature.

The results generalize previous rigidity theorems to broader classes of manifolds.

Abstract

The lens data of a Riemannian manifold with boundary is the collection of lengths of geodesics with endpoints on the boundary together with their incoming and outgoing vectors. We show that negatively-curved Riemannian manifolds with strictly convex boundary are locally lens rigid in the following sense: if $g_0$ is such a metric, then any metric $g$ sufficiently close to $g_0$ and with same lens data is isometric to $g_0$, up to a boundary-preserving diffeomorphism. More generally, we consider the same problem for a wider class of metrics with strictly convex boundary, called metrics of Anosov type. We prove that the same rigidity result holds within that class in dimension $2$ and in any dimension, further assuming that the curvature is non-positive.