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Boundary and lens rigidity, tensor tomography and analytic microlocal analysis

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Algebraic Analysis of Differential Equations

Abstract

The boundary rigidity problem consists of determining a compact, Riemannian manifold with boundary, up to isometry, by knowing the boundary distance function between boundary points. Lens rigidity consists of determining the manifold, by knowing the scattering relation which measures, besides the travel times, the point and direction of exit of a geodesic from the manifold if one knows its point and direction of entrance. Tensor tomography is the linearization of boundary rigidity and length rigidity. It consists of determining a symmetric tensor of order two from its integral along geodesics. In this paper we survey some recent results obtained on these problems using methods from microlocal analysis, in particular analytic microlocal analysis. Although we use the distribution version of analytic microlocal analysis, many of the ideas were based on the pioneer work of the Sato school of microlocal analysis of which Professor Kawai was a very important member.

The first author was supported in part by NSF Grant DMS-0400869.

The second author was supported in part by NSF Grant DMS-0245414.

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Author information

Authors and Affiliations

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

    Plamen Stefanov

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

    Gunther Uhlmann

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  1. Plamen Stefanov
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  2. Gunther Uhlmann
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Kinki University, Higashi-Osaka, 577-8502, Japan

    Takashi Aoki

  2. Department of Mathematics, Ochanomizu University, Tokyo, 112-8610, Japan

    Hideyuki Majima

  3. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Yoshitsugu Takei

  4. Faculty of Economics, Keio University, Yokohama, 223-8521, Japan

    Nobuyuki Tose

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Stefanov, P., Uhlmann, G. (2008). Boundary and lens rigidity, tensor tomography and analytic microlocal analysis. In: Aoki, T., Majima, H., Takei, Y., Tose, N. (eds) Algebraic Analysis of Differential Equations. Springer, Tokyo. https://doi.org/10.1007/978-4-431-73240-2_23

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  • DOI: https://doi.org/10.1007/978-4-431-73240-2_23

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  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-73239-6

  • Online ISBN: 978-4-431-73240-2

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