Inverting the local geodesic X-ray transform on tensors

TL;DR

This paper establishes local invertibility and stability of the geodesic X-ray transform on tensor fields near a convex boundary, providing an inversion formula and global results under certain foliation conditions.

Contribution

It proves new local and global invertibility results for the geodesic X-ray transform on tensors, including an inversion formula and conditions for lens rigidity.

Findings

Local invertibility and stability near convex boundary points

Global invertibility under foliation conditions

Applicable to manifolds with no focal points or conjugate points

Abstract

We prove the local invertibility, up to potential fields, and stability of the geodesic X-ray transform on tensor fields of order 1 and 2 near a strictly convex boundary point, on manifolds with boundary of dimension n>=3. We also present an inversion formula. Under the condition that the manifold can be foliated with a continuous family of strictly convex surfaces, we prove a global result which also implies a lens rigidity result near such a metric. The class of manifolds satisfying the foliation condition includes manifolds with no focal points, and does not exclude existence of conjugate points.