On mixed and transverse ray transforms on orientable surfaces

TL;DR

This paper studies a class of mixing ray transforms on orientable surfaces, establishing their kernel and stability properties through algebraic methods, and extending previous results to more general geometries.

Contribution

It introduces a unified approach to analyze mixing ray transforms, including the light ray transform, on orientable surfaces, broadening the scope of prior work.

Findings

Kernel characterization of mixing ray transforms

Stability results for these transforms

Extension to orientable surfaces with solenoidally injective geodesic ray transform

Abstract

The geodesic ray transform, the mixed ray transform and the transverse ray transform of a tensor field on a manifold can all be seen as what we call mixing ray transforms, compositions of the geodesic ray transform and an invertible linear map on tensor fields. We show that the characterization of the kernel and the stability of a mixing ray transform can be reduced to the same properties of any other mixing ray transform. Our approach applies to various geometries and ray transforms, including the light ray transform. In particular, we extend studies in de Hoop--Saksala--Zhai (2019) from compact simple surfaces to orientable surfaces with solenoidally injective geodesic ray transform. Our proofs are based on algebraic arguments.