Integral geometry of tensor fields on a class of non-simple Riemannian manifolds

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