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Boundary and lens rigidity, tensor tomography and analytic microlocal analysis

In: Algebraic Analysis of Differential Equations

Author

Listed:
  • Plamen Stefanov

    (Purdue University, Department of Mathematics)

  • Gunther Uhlmann

    (University of Washington, Department of Mathematics)

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.

Suggested Citation

  • Plamen Stefanov & Gunther Uhlmann, 2008. "Boundary and lens rigidity, tensor tomography and analytic microlocal analysis," Springer Books, in: Takashi Aoki & Hideyuki Majima & Yoshitsugu Takei & Nobuyuki Tose (ed.), Algebraic Analysis of Differential Equations, pages 275-293, Springer.
    • Handle: RePEc:spr:sprchp:978-4-431-73240-2_23
    DOI: 10.1007/978-4-431-73240-2_23
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