The attenuated ray transforms on Gaussian thermostats with negative curvature

TL;DR

This paper investigates the injectivity and reconstruction properties of attenuated ray transforms on Gaussian thermostats with negative curvature, extending results to surfaces with boundary and closed surfaces under certain conditions.

Contribution

It establishes injectivity of the thermostat ray transform with attenuation on negatively curved surfaces and shows how to recover connections and Higgs fields from boundary data.

Findings

Injectivity of the ray transform with attenuation on surfaces with boundary.

Unique determination of connections and Higgs fields from boundary parallel transport.

Results for closed surfaces with conditions on the connection and Higgs fields.

Abstract

In the present paper we consider a Gaussian thermostat on a compact Riemannian surface with negative thermostat curvature. In the case of surfaces with boundary, we show that the thermostat ray transform with attenuation given by a general connection and Higgs field is injective, modulo the natural obstruction, for tensors. We also prove that the connection and Higgs field can be determined, up to a gauge transformation, from the knowledge of the parallel transport between boundary points along all possible thermostat geodesics. In the case of closed surfaces, we obtain similar results for the ray transform with some additional conditions on the connection. Under the same condition, we study connections and Higgs fields whose parallel transport along periodic thermostat geodesics coincides with the ones for the flat connection.