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

TL;DR

This paper establishes the injectivity of the geodesic X-ray transform on simple Riemannian manifolds with low regularity metrics, extending previous results to manifolds with $C^{1,1}$ regularity and redefining simplicity accordingly.

Contribution

It proves injectivity of the X-ray transform on manifolds with $C^{1,1}$ metrics and introduces a compatible notion of simplicity for rough geometries.

Findings

Injectivity of the X-ray transform on scalar functions and one-forms on $C^{1,1}$ manifolds.

Redefinition of simplicity compatible with low-regularity metrics.

Development of calculus of differential operators on non-smooth structures.

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 $g \in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $C^{1,1}$-regularity is optimal on the H\"older 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.