A1 Journal article (refereed)
The geodesic X-ray transform with matrix weights (2019)

Paternain, G. B., Salo, M., Uhlmann, G., & Zhou, H. (2019). The geodesic X-ray transform with matrix weights. American Journal of Mathematics, 141(6), 1707-1750. https://doi.org/10.1353/ajm.2019.0045

JYU authors or editors

Salo, Mikko

Publication details

All authors or editors: Paternain, Gabriel B.; Salo, Mikko; Uhlmann, Gunther; Zhou, Hanming

Journal or series: American Journal of Mathematics

ISSN: 0002-9327

eISSN: 1080-6377

Publication year: 2019

Volume: 141

Issue number: 6

Pages range: 1707-1750

Publisher: Johns Hopkins University Press

Publication country: United States

Publication language: English

DOI: https://doi.org/10.1353/ajm.2019.0045

Publication open access: Not open

Publication channel open access:

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/67886

Publication is parallel published: https://arxiv.org/abs/1605.07894

Abstract

Consider a compact Riemannian manifold of dimension ≥ 3 with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-forms. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo gauge transformations. The proofs involve a reduction to a local result showing that the geodesic X-ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary, and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions. As a somewhat striking corollary, we show that these integral geometry problems can be solved on strictly convex manifolds of dimension ≥ 3 having nonnegative sectional curvature (similar results were known earlier in negative sectional curvature). We also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography.

Keywords: inverse problems; integral equations; manifolds (mathematics)

Fields of science: