Integral Geometry of Tensor Fields on a Class of Non-Simple Riemannian Manifolds

We study the geodesic X-ray transform IΓ of tensor fields on a compact Riemannian manifold M with non-necessarily convex boundary and with possible conjugate points. We assume that IΓ is known for geodesics belonging to an open set Γ with endpoints on the boundary. We prove generic s-injectivity and a stability estimate under some topological assumptions and under the condition that for any (x, ξ) ∈ T*M, there is a geodesic in Γ through x normal to ξ without conjugate points.