Local Lens Rigidity with Incomplete Data for a Class of Non-simple Riemannian Manifolds

Let σ be the scattering relation on a compact Riemannian manifold M with nonnecessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let l be the length of that geodesic ray. We study the question of whether the metric g is uniquely determined, up to an isometry, by knowledge of σ and l restricted on some subset D. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from D covers T ∗M ; and that those geodesics have no conjugate points. Under an additional topological assumption, we prove that σ and l restricted to D uniquely recover an isometric copy of g locally near generic metrics, and in particular, near real analytic ones.