The X-Ray Transform for Connections in Negative Curvature

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• 作者：Colin Guillarmou ; Gabriel P. Paternain…
• 刊名：Communications in Mathematical Physics
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：343
• 期：1
• 页码：83-127
• 全文大小：901 KB
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• 作者单位：Colin Guillarmou (1)
  Gabriel P. Paternain (2)
  Mikko Salo (3)
  Gunther Uhlmann (4) (5) (6)
  
  1. DMA, Ecole Normale Superieure, Paris, France
  2. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, UK
  3. Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland
  4. University of Washington, Seattle, USA
  5. University of Helsinki, Helsinki, Finland
  6. Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Quantum Physics
  Quantum Computing, Information and Physics
  Complexity
  Statistical Physics
  Relativity and Cosmology
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0916

文摘

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.