Broken ray tensor tomography with one reflecting obstacle

TL;DR

This paper characterizes tensor fields that integrate to zero over broken rays in a geometry with an obstacle, revealing they are symmetrized derivatives of lower tensors satisfying boundary conditions, with new proofs applicable even without reflections.

Contribution

It provides a novel characterization of tensor fields integrating to zero over broken rays in curved geometries with obstacles, including new proofs and results for scalar fields in higher dimensions.

Findings

Tensor fields integrating to zero are symmetrized derivatives of lower tensors.

The results hold in non-positive curvature geometries with convex obstacles.

New proofs extend to cases without reflections, even for scalars in higher dimensions.

Abstract

We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional curvature and a strictly convex obstacle in any dimension. We give two proofs, both of which contain new features also in the absence of reflections. The result is new even for scalars in dimensions above two.