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

Abstract

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 ∈ C1,1. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This C1,1-regularity is optimal on the Hölder 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.