Integral geometry problem for nontrapping manifolds

We consider the integral geometry problem of restoring a tensor field on a manifold with boundary from its integrals over geodesics running between boundary points. For nontrapping manifolds with a certain upper curvature bound, we prove that a tensor field, integrating to zero over geodesics between boundary points, is potential. For functions and 1-forms, this is shown to be true for arbitrary nontrapping manifolds with no conjugate points. As a consequence, we also establish deformation boundary rigidity for strongly geodesic minimizing manifolds with a certain upper curvature bound.