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The X-Ray Transform for Connections in Negative Curvature

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Abstract

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of the attenuated ray transform on tensor fields with values in a Hermitian bundle (i.e., vector valued case). We also show that a connection and Higgs field on a Hermitian bundle are determined up to gauge by the knowledge of the parallel transport between boundary points along all possible geodesics. The main tools are an energy identity, the Pestov identity with a unitary connection, which is presented in a general form, and a precise analysis of the singularities of solutions of transport equations when there are trapped geodesics. In the case of closed manifolds, we obtain similar results modulo the obstruction given by twisted conformal Killing tensors, and we also study this obstruction.

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Authors and Affiliations

  1. DMA, Ecole Normale Superieure, Paris, France

    Colin Guillarmou

  2. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, UK

    Gabriel P. Paternain

  3. Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

    Mikko Salo

  4. University of Washington, Seattle, USA

    Gunther Uhlmann

  5. University of Helsinki, Helsinki, Finland

    Gunther Uhlmann

  6. Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

    Gunther Uhlmann

Authors
  1. Colin Guillarmou
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  2. Gabriel P. Paternain
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  3. Mikko Salo
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  4. Gunther Uhlmann
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Correspondence to Gabriel P. Paternain.

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Communicated by S. Zelditch

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Guillarmou, C., Paternain, G.P., Salo, M. et al. The X-Ray Transform for Connections in Negative Curvature. Commun. Math. Phys. 343, 83–127 (2016). https://doi.org/10.1007/s00220-015-2510-x

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  • DOI: https://doi.org/10.1007/s00220-015-2510-x

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