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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-tekijät tai -toimittajat

Salo, Mikko

Julkaisun tiedot

Julkaisun kaikki tekijät tai toimittajat: Paternain, Gabriel B.; Salo, Mikko; Uhlmann, Gunther; Zhou, Hanming

Lehti tai sarja: American Journal of Mathematics

ISSN: 0002-9327

eISSN: 1080-6377

Julkaisuvuosi: 2019

Volyymi: 141

Lehden numero: 6

Artikkelin sivunumerot: 1707-1750

Kustantaja: Johns Hopkins University Press

Julkaisumaa: Yhdysvallat (USA)

Julkaisun kieli: englanti

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

Julkaisun avoin saatavuus: Ei avoin

Julkaisukanavan avoin saatavuus:

Julkaisu on rinnakkaistallennettu (JYX): https://jyx.jyu.fi/handle/123456789/67886

Julkaisu on rinnakkaistallennettu: https://arxiv.org/abs/1605.07894

Tiivistelmä

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.

YSO-asiasanat: inversio-ongelmat; integraaliyhtälöt; monistot

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