A support theorem for the geodesic ray transform of symmetric tensor fields

TL;DR

This paper proves a support theorem for the geodesic ray transform of symmetric tensor fields on real-analytic simple manifolds, using microlocal analysis to establish conditions under which the tensor field is a symmetric differential of a vector field.

Contribution

It establishes a support theorem for the geodesic ray transform of symmetric 2-tensor fields on real-analytic simple manifolds, extending previous results with microlocal techniques.

Findings

Support theorem for symmetric tensor fields established

Conditions for tensor fields to be symmetric differentials derived

Microlocal analysis applied to geodesic ray transform

Abstract

Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown that there exists a vector field $v$ satisfying $f=dv$ on the set of points lying on these geodesics and $v=0$ on the intersection of this set with the boundary $\PD M$ of the manifold $M$. Using this result, a Helgason's type of a support theorem for the geodesic ray transform is proven. The approach is based on analytic microlocal techniques.