Lens rigidity for manifolds with hyperbolic trapped set

الأصل · EN

For a Riemannian manifold (M,g) with strictly convex boundary ∂ M, the lens data consists in the set of lengths of geodesics γ with endpoints on ∂ M, together with their endpoints (x-,x+)∈ ∂ M× ∂ M and tangent exit vectors (v-,v+)∈ Tₓ₋ M× Tₓ₊ M. We show deformation lens rigidity for a large class of manifolds which includes all manifolds with negative curvature and strictly convex boundary, possibly with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension 2, we prove that the set of endpoints and exit vectors of geodesics (ie. the scattering data) determines the topology and the conformal class of the surface.