Broken Ray Transform for Twisted Geodesics on Surfaces with a Reflecting Obstacle

We prove a uniqueness result for the broken ray transform acting on the sums of functions and 1-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have nonpositive curvature. The broken rays are generated from the twisted geodesic flows by the law of reflection on the boundary of a suitably convex obstacle. Our work generalizes recent results for the broken geodesic ray transform on surfaces to more general families of curves including the magnetic flows and Gaussian thermostats.

Description

Acknowledgements: M.K. would like to thank Mikko Salo for suggesting research on the magnetic broken ray transforms and helpful discussions. J.R. thanks Gabriel P. Paternain for many helpful discussions related to this work. We thank an anonymous referee for many helpful comments that improved the final manuscript. S.R.J. and J.R. would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, for support and hospitality during Rich and Nonlinear Tomography - a multidisciplinary approach in 2023 where part of this work was done (supported by EPSRC Grant Number EP/R014604/1). The work of S.R.J. and J.R. was supported by the Research Council of Finland through the Flagship of Advanced Mathematics for Sensing, Imaging and Modeling (decision number 359183). S.R.J. acknowledges the Prime Minister’s Research Fellowship (PMRF) from the Government of India for his PhD work. M.K. was supported by MATRICS grant (MTR/2019/001349) of SERB. J.R. was supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters.

Funder: Väisälän Rahasto; doi: http://dx.doi.org/10.13039/501100016032

Funder: LUT University (previously Lappeenranta University of Technology (LUT))

Keywords

37E35, Magnetic flows, Geodesic ray transform, 58C99, Inverse problems, Broken rays, Gaussian thermostats, 44A12

Journal Title

The Journal of Geometric Analysis

Journal ISSN

1050-6926
1559-002X

Volume Title

34

Publisher

Springer US

Publisher DOI

https://doi.org/10.1007/s12220-024-01653-2

Rights and licensing

Except where otherwised noted, this item's license is described as http://creativecommons.org/licenses/by/4.0/

Sponsorship

Academy of Finland (359183, 359183)

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