Inverting the local geodesic ray transform of higher rank tensors

TL;DR

This paper proves local invertibility of the geodesic ray transform for rank four tensors on certain Riemannian manifolds, with implications for elastic wave tomography and potential for generalization to higher ranks.

Contribution

It establishes local invertibility of the geodesic ray transform for rank four tensors near a boundary point, extending to arbitrary rank under certain conditions.

Findings

Local invertibility of the geodesic ray transform for rank four tensors.

Global invertibility under foliation conditions.

Adaptability of the proof to higher rank tensors.

Abstract

Consider a Riemannian manifold in dimension $n\geq 3$ with strictly convex boundary. We prove the local invertibility, up to potential fields, of the geodesic ray transform on tensor fields of rank four near a boundary point. This problem is closely related with elastic \textit{qP}-wave tomography. Under the condition that the manifold can be foliated with a continuous family of strictly convex hypersurfaces, the local invertibility implies a global result. One can straightforwardedly adapt the proof to show similar results for tensor fields of arbitrary rank.