LOCAL LENS RIGIDITY FOR MANIFOLDS OF ANOSOV TYPE

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 the 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 nonpositive.