Local lens rigidity for manifolds of Anosov type

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 g0 is such a metric, then any metric g sufficiently close to g0 and with same lens data is isometric to g0, 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.