Pestov identities and x-ray tomography on manifolds of low regularity

We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds (M, g) with g is an element of C-1,C-1. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This C-1,C-1-regularity is optimal on the Holder scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure.