Support theorem for the transverse ray transform of tensor fields of rank 2

TL;DR

This paper proves a support theorem for the transverse ray transform of rank-2 tensor fields on simple, real analytic Riemannian manifolds, establishing conditions under which the tensor field's support can be determined from its transform.

Contribution

It introduces a support theorem for the transverse ray transform of symmetric rank-2 tensor fields on simple manifolds, extending previous results to this specific setting.

Findings

Support theorem established for tensor fields of rank 2

Support of tensor field vanishes where the transform vanishes

Results apply to simple, real analytic Riemannian manifolds

Abstract

Let (M, g) be a simple, real analytic, Riemannian manifold with boundary and of dimension n>=3. In this work, we prove a support theorem for the transverse ray transform of tensor fields of rank 2 defined over such manifolds. More specifically, given a symmetric tensor field f of rank 2, we show that if the transverse ray transform of f vanishes over an appropriate open set of maximal geodesics of M , then the support of f vanishes on the points of M that lie on the union of the aforementioned open set of geodesics.