Communications in Analysis and Geometry

Volume 30 (2022)

Number 6

Broken ray tensor tomography with one reflecting obstacle

Pages: 1269 – 1300



Joonas Ilmavirta (Department of Mathematics and Statistics, University of Jyväskylä, Finland)

Gabriel P. Paternain (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom)


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.

Much of the research leading to this paper was completed during the inverse problems program at Institut Henri Poincaré in 2015 and during the first author’s visits to Cambridge. The authors thank CNRS and IHP for support and hospitality. J.I. was supported by the Academy of Finland (decision 295853) and an encouragement grant from the Emil Aaltonen foundation, and he is grateful for all the hospitality at Cambridge. G.P.P. was supported by an EPSRC grant (EP/R001898/1). We are grateful to the anonymous referees for useful feedback.

Received 19 May 2018

Accepted 12 January 2020

Published 26 April 2023