Boundary and lens rigidity for non-convex manifolds

TL;DR

This paper investigates boundary and lens rigidity problems on non-convex Riemannian surfaces, establishing rigidity results and X-ray transform injectivity under various geometric conditions without boundary convexity.

Contribution

It extends boundary and lens rigidity results to non-convex domains and proves X-ray transform injectivity in more general non-trapping and non-convex settings.

Findings

Rigidity holds for simply connected compact surfaces without conjugate points.

Lens rigidity is established for non-trapping surfaces with no conjugate points.

Injectivity of the X-ray transform is proven in various non-convex boundary scenarios.

Abstract

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the more general class of non-trapping compact Riemannian surfaces with no conjugate points, we show lens rigidity. We also prove the injectivity of the X-ray transform on tensors in a variety of settings with non-convex boundary and, in some situations, allowing a non-empty trapped set.