The geodesic X-ray transform with matrix weights
Paternain, Gabriel B.; Salo, Mikko; Uhlmann, Gunther; Zhou, Hanming (2019). The geodesic X-ray transform with matrix weights. American Journal of Mathematics, 141 (6), 1707-1750. DOI: 10.1353/ajm.2019.0045
Published inAmerican Journal of Mathematics
DisciplineInversio-ongelmien huippuyksikköMatematiikkaCentre of Excellence in Inverse ProblemsMathematics
© 2019 by Johns Hopkins University Press
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. ...
PublisherJohns Hopkins University Press
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Related funder(s)Academy of Finland; European Commission
Funding program(s)Academy Project, AoF; FP7 (EU's 7th Framework Programme); Centre of Excellence, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.