Please use this identifier to cite or link to this item: https://hdl.handle.net/1783.1/100696
The geodesic X-ray transform with matrix weights
| Author |
Paternain, Gabriel P.
Salo, Mikko Uhlmann, Gunther Alberto Arancibia Zhou, Hanming |
|---|---|
| Issue Date | 2019 |
| Source | American Journal of Mathematics, v. 141, (6), December 2019, p. 1707-1750 |
| 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. |
| DOI | 10.1353/ajm.2019.0045 |
| ISSN | 0002-9327 |
| Language | English |
| Type | Article |
| Access |
View Open Access
Version View full-text via Browzine View details via DOI View details via Web of Science View details via Scopus |