Ray transform of symmetric tensor fields on Riemannian manifolds with conjugate points

TL;DR

This paper analyzes the microlocal properties of the geodesic ray transform of symmetric tensor fields on 2D Riemannian manifolds with boundary, including conjugate points, and constructs a parametrix to recover tensor components.

Contribution

It explicitly computes the principal symbols of the normal operator's components and establishes a cancellation of singularities specific to 2D tensor field transforms.

Findings

Principal symbol of the normal operator components is explicitly computed.

A parametrix for the solenoidal component of tensor fields is constructed.

A cancellation of singularities result is proved, unique to 2D cases.

Abstract

In this article, we study the microlocal properties of the geodesic ray transform of symmetric $m$-tensor fields on 2-dimensional Riemannian manifolds with boundary allowing the possibility of conjugate points. As is known from an earlier work on the geodesic ray transform of functions in the presence of conjugate points, the normal operator can be decomposed into a sum of a pseudodifferential operator ($\Psi$DO) and a finite number of Fourier integral operators (FIOs) under the assumption of no singular conjugate pairs along geodesics, which always holds in 2-dimensions. In this work, we use the method of stationary phase to explicitly compute the principal symbol of the $\Psi$DO and each of the FIO components of the normal operator acting on symmetric $m$-tensor fields. Next, we construct a parametrix recovering the solenoidal component of the tensor fields modulo FIOs, and prove a cancellation of singularities result, similar to an earlier result of Monard, Stefanov and Uhlmann for the case of geodesic ray transform of functions in 2-dimensions. We point out that this type of cancellation result is only possible in the 2-dimensional case.