The geodesic ray transform on spherically symmetric reversible Finsler manifolds

TL;DR

This paper proves the injectivity of the geodesic ray transform on scalar functions in spherically symmetric reversible Finsler manifolds satisfying a Herglotz condition, with implications for boundary rigidity and travel time tomography.

Contribution

It establishes the injectivity of the geodesic ray transform on such Finsler manifolds using Fourier series and Abel transform techniques, extending previous results to this class of manifolds.

Findings

Injectivity of the geodesic ray transform on scalar functions.

Reduction of the problem to invertibility of Abel transforms.

Applications to boundary rigidity and elastic travel time tomography.

Abstract

We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.