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The geodesic X-ray transform with matrix weights

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Paternain, GP 
Salo, M 
Uhlmann, G 

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 non-negative 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.

Description

Keywords

math.DG, math.DG, math-ph, math.AP, math.MP

Journal Title

American Journal of Mathematics

Conference Name

Journal ISSN

0002-9327
1080-6377

Volume Title

141

Publisher

Project MUSE
Sponsorship
Engineering and Physical Sciences Research Council (EP/M023842/1)