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The X-Ray Transform for Connections in Negative Curvature


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Authors

Guillarmou, Colin 
Paternain, Gabriel P 
Salo, Mikko 
Uhlmann, Gunther 

Abstract

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of the attenuated ray transform on tensor fields with values in a Hermitian bundle (i.e. vector valued case). We also show that a connection and Higgs field on a Hermitian bundle are determined up to gauge by the knowledge of the parallel transport between boundary points along all possible geodesics. The main tools are an energy identity, the Pestov identity with a unitary connection, which is presented in a general form, and a precise analysis of the singularities of solutions of transport equations when there are trapped geodesics. In the case of closed manifolds, we obtain similar results modulo the obstruction given by twisted conformal Killing tensors, and we also study this obstruction.

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Keywords

math.AP, math.AP, math.DG, math.DS

Journal Title

COMMUNICATIONS IN MATHEMATICAL PHYSICS

Conference Name

Journal ISSN

0010-3616
1432-0916

Volume Title

343

Publisher

Springer Science and Business Media LLC
Sponsorship
C.G. was partially supported by grants ANR-13-BS01-0007- 01 and ANR-13-JS01-0006. M.S. was supported in part by the Academy of Finland (Centre of Excellence in Inverse Problems Research) and an ERC Starting Grant (grant agreement no 307023). G.U. was partly supported by NSF and a Simons Fellowship.