Lens rigidity for manifolds with hyperbolic trapped sets

Abstract:

For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consist of the set of lengths of geodesics $\gamma$ with end points on $\partial M$, together with their end points $(x_-,x_+)\in \partial M\times \partial M$ and tangent exit vectors $(v_-,v_+)\in T_{x_-} M\times T_{x_+} M$. We show deformation lens rigidity for such manifolds with a hyperbolic trapped set and no conjugate points. This class contains all manifolds with negative curvature and strictly convex boundary, including those with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension $2$, we prove that the set of end points and exit vectors of geodesics (i.e., the scattering data) determines the Riemann surface up to conformal diffeomorphism.