Inverting the local geodesic X-ray transform on tensors

We prove the local invertibility, up to potential fields, and stability of the geodesic X-ray transform on tensor fields of order 1 and 2 near a strictly convex boundary point, on manifolds with boundary of dimension n ≥ 3. We also present an inversion formula. Under the condition that the manifold...

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Bibliographic Details
Published inJournal d'analyse mathématique (Jerusalem) Vol. 136; no. 1; pp. 151 - 208
Main Authors Stefanov, Plamen, Uhlmann, Gunther, Vasy, András
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.10.2018
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Analysis
Conjugate points
Dynamical Systems and Ergodic Theory
Functional Analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Partial Differential Equations
Potential fields
Tensors
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Summary:We prove the local invertibility, up to potential fields, and stability of the geodesic X-ray transform on tensor fields of order 1 and 2 near a strictly convex boundary point, on manifolds with boundary of dimension n ≥ 3. We also present an inversion formula. Under the condition that the manifold can be foliated with a continuous family of strictly convex surfaces, we prove a global result which also implies a lens rigidity result near such a metric. The class of manifolds satisfying the foliation condition includes manifolds with no focal points, and does not exclude existence of conjugate points.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-018-0058-3