Notes on tensor tomography on Riemannian manifolds with boundaries

TL;DR

This paper reviews key proofs establishing the s-injectivity of tensor tomography on compact Riemannian manifolds with boundaries, under specific geometric conditions, consolidating prior significant results.

Contribution

It summarizes and clarifies the proofs of s-injectivity for tensor tomography on Riemannian manifolds, highlighting the conditions under which these results hold.

Findings

Proofs of s-injectivity under geometric assumptions

Consolidation of prior results from 2006 and 2015

Clarification of conditions for tensor tomography on manifolds

Abstract

We summarize the proofs for the s-injectivity of the tensor tomography problem on compact Riemannian manifolds with boundaries in [Dairbekov, Inverse Problems, 22: 431, 2006] and [Paternain-Salo-Uhlmann, Math. Ann., 363: 305-362, 2015] under certain geometric assumptions.