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Inverting the local geodesic X-ray transform on tensors

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907 1395, USA

    Plamen Stefanov

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195 4350, USA

    Gunther Uhlmann

  3. University of Helsinki, and Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong, China

    Gunther Uhlmann

  4. Department of Mathematics, Stanford University, Stanford, CA, 94305 2125, USA

    András Vasy

Authors
  1. Plamen Stefanov
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  2. Gunther Uhlmann
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  3. András Vasy
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Corresponding author

Correspondence to András Vasy.

Additional information

Partially supported by the National Science Foundation under grant DMS-1301646

Partially supported by the National Science Foundation under grants CMG-1025259 and DMS-1265958

Partially supported by the National Science Foundation under grants CMG-1025259 and DMS-1361432

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Stefanov, P., Uhlmann, G. & Vasy, A. Inverting the local geodesic X-ray transform on tensors. JAMA 136, 151–208 (2018). https://doi.org/10.1007/s11854-018-0058-3

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  • DOI: https://doi.org/10.1007/s11854-018-0058-3

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