The Attenuated Magnetic Ray Transform on Surfaces

TL;DR

This paper extends the injectivity and uniqueness results of the attenuated geodesic ray transform to magnetic geodesics on surfaces, with applications to tensor tomography, advancing understanding in geometric inverse problems.

Contribution

It generalizes previous results to magnetic geodesics and demonstrates how the scattering relation determines the connection and Higgs field up to gauge transformations.

Findings

Injectivity of the magnetic ray transform on functions and 1-forms.

Determination of connection and Higgs field from scattering relation.

Application to tensor tomography in magnetic settings.

Abstract

It has been shown in [Pa1] that on a simple, compact Riemannian 2-manifold the attenuated geodesic ray transform, with attenuation given by a connection and Higgs field, is injective on functions and 1-forms modulo the natural obstruction. Furthermore, the scattering relation determines the connection and Higgs field modulo a gauge transformation. We extend the results obtained therein to the case of magnetic geodesics. In addition, we provide an application to tensor tomography in the magnetic setting, along the lines of [Pa2].