LENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET
Résumé
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 +) ∈ T x− M × T x+ M. We show deformation lens rigidity for manifolds with hyperbolic trapped set and no conjugate points, a class which 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 endpoints and exit vectors of geodesics (ie. the scattering data) determines the Riemann surface up to conformal diffeomorphism.
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