The attenuated geodesic ray transform on tensors: generic injectivity and stability

TL;DR

This paper proves injectivity and stability of the attenuated geodesic ray transform on tensors for generic simple metrics and attenuations, with potential extensions to non-simple manifolds, advancing inverse problems in geometric analysis.

Contribution

It establishes new injectivity and stability results for the attenuated geodesic ray transform on tensors on simple Riemannian manifolds, including real analytic cases.

Findings

Injectivity and stability proven for a class of generic simple metrics and attenuations.

Methods can be extended to certain non-simple manifolds.

Results contribute to inverse problems in geometric analysis.

Abstract

We consider the attenuated geodesic ray transform defined on pairs of symmetric $2$-tensors and $1$-forms on a simple Riemannian manifold. We prove injectivity and stability results for a class of generic simple metrics and attenuations containing real analytic ones. In fact, methods used in this paper can be modified to generalize our results for a class of non-simple manifolds similar to Stefanov-Uhlmann [American Journal of Mathematics, {\bf 130} (1):239--268 (2008)].