Advertisement

Boundary and lens rigidity, tensor tomography and analytic microlocal analysis

  • Plamen Stefanov
  • Gunther Uhlmann
Chapter

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.

Keywords

Symmetric Tensor Stability Estimate Conjugate Point Compact Riemannian Manifold Boundary Rigidity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. Alexandrova, Semi-Classical wavefront set and Fourier integral operators, to appear in Can. J. Math. Google Scholar
  2. 2.
    _____, Structure of the Semi-Classical Amplitude for General Scattering Relations, Comm. PDE 30(2005), 1505–1535.Google Scholar
  3. 3.
    M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its boundary spectral data (BC-method), Comm. PDE 17(1992), 767–804.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    I. N. Bernstein and M. L. Gerver, Conditions on distinguishability of metrics by hodographs, Methods and Algorithms of Interpretation of Seismological Information, Computerized Seismology 13, Nauka, Moscow, 50–73 (in Russian).Google Scholar
  5. 5.
    G. Besson, G. Courtois, and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictment négative, Geom. Funct. Anal., 5(1995), 731–799.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    G. Beylkin, Stability and uniqueness of the solution of the inverse kinematic problem in the multidimensional case, J. Soviet Math. 21(1983), 251–254.CrossRefGoogle Scholar
  7. 7.
    D. Burago and S. Ivanov, Boundary rigidity and filling volume minimality of metrics close to a flat one, manuscript, 2005.Google Scholar
  8. 8.
    K. C. Creager, Anisotropy of the inner core from differential travel times of the phases PKP and PKIPK, Nature, 356(1992), 309–314.CrossRefGoogle Scholar
  9. 9.
    C. Croke, Rigidity for surfaces of non-positive curvature, Comment. Math. Helv., 65(1990), 150–169.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    _____, Rigidity and the distance between boundary points, J. Differential Geom., 33(1991), no. 2, 445–464.MATHMathSciNetGoogle Scholar
  11. 11.
    C. Croke, N. Dairbekov, V. Sharafutdinov, Local boundary rigidity of a compact Riemannian manifold with curvature bounded above, Trans. Amer. Math. Soc. 352(2000), no. 9, 3937–3956.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    C. Croke and B. Kleiner, Conjugacy and Rigidity for Manifolds with a Parallel Vector Field, J. Diff. Geom. 39(1994), 659–680.MATHMathSciNetGoogle Scholar
  13. 13.
    M. Gromov, Filling Riemannian manifolds, J. Diff. Geometry 18(1983), no. 1, 1–148.MATHMathSciNetGoogle Scholar
  14. 14.
    V. Guillemin, Sojourn times and asymptotic properties of the scattering matrix, Proceedings of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on Algebraic Analysis (Kyoto Univ., Kyoto, 1976). Publ. Res. Inst. Math. Sci. 12(1976/77), supplement, 69–88.Google Scholar
  15. 15.
    G. Herglotz, Uber die Elastizitaet der Erde bei Beruecksichtigung ihrer variablen Dichte, Zeitschr. fur Math. Phys., 52(1905), 275–299.Google Scholar
  16. 16.
    C. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón Problem with partial data, to appear in Ann. Math. Google Scholar
  17. 17.
    M. Lassas, V. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Math. Ann., 325(2003), 767–793.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65(1981), 71–83.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (Russian), Dokl. Akad. Nauk SSSR 232(1977), no. 1, 32–35.MathSciNetGoogle Scholar
  20. 20.
    R. G. Mukhometov, On a problem of reconstructing Riemannian metrics Siberian Math. J. 22(1982), no. 3, 420–433.CrossRefMathSciNetGoogle Scholar
  21. 21.
    R. G. Mukhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in an n-dimensional space (Russian), Dokl. Akad. Nauk SSSR 243(1978), no. 1, 41–44.MathSciNetGoogle Scholar
  22. 22.
    J. P. Otal, Sur les longuer des géodésiques d’une métrique a courbure négative dans le disque, Comment. Math. Helv. 65(1990), 334–347.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    L. Pestov, V. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature (Russian) Sibirsk. Mat. Zh. 29(1988), no. 3, 114–130; translation in Siberian Math. J. 29(1988), no. 3, 427–441.MATHMathSciNetGoogle Scholar
  24. 24.
    L. Pestov and G. Uhlmann, Two dimensional simple compact manifolds with boundary are boundary rigid, Annals of Math. 161(2005), 1089–1106.MathSciNetCrossRefGoogle Scholar
  25. 25.
    V. Sharafutdinov, Integral geometry of tensor fields, VSP, Utrech, the Netherlands, 1994.Google Scholar
  26. 26.
    V. Sharafutdinov, M. Skokan, and G. Uhlmann, Regularity of ghosts in tensor tomography, Journal of Geometric Analysis textbf15(2005), 517–560.Google Scholar
  27. 27.
    V. Sharafutdinov and G. Uhlmann, On deformation boundary rigidity and spectral rigidity for Riemannian surfaces with no focal points, Journal of Differential Geometry, 56 (2001), 93–110.MathSciNetGoogle Scholar
  28. 28.
    J. Sjöstrand, Singularités analytiques microlocales, Astérique 95(1982), 1–166.MATHGoogle Scholar
  29. 29.
    P. Stefanov and G. Uhlmann, Rigidity for metrics with the same lengths of geodesics, Math. Res. Lett. 5(1998), 83–96.MATHMathSciNetGoogle Scholar
  30. 30.
    _____, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J. 123(2004), 445–467.MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    _____, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, IMRN 17(2005), 1047–1061.CrossRefMathSciNetGoogle Scholar
  32. 32.
    _____, Boundary rigidity and stability for generic simple metrics, Journal Amer. Math. Soc. 18(2005), 975–1003.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    _____, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds, arXiv:math.DG/0601178, to appear in Amer. J. Math.Google Scholar
  34. 34.
    _____, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, in progress.Google Scholar
  35. 35.
    D. Tataru, Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem, Comm. P.D.E. 20(1995), 855–884.MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1. Pseudodifferential Operators. The University Series in Mathematics, Plenum Press, New York-London, 1980.MATHGoogle Scholar
  37. 37.
    J. Wang, Stability for the reconstruction of a Riemannian metric by boundary measurements, Inverse Probl. 15(1999), 1177–1192.MATHCrossRefGoogle Scholar
  38. 38.
    E. Wiechert E and K. Zoeppritz, Uber Erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss, Goettingen 4(1907), 415–549.Google Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  • Plamen Stefanov
    • 1
  • Gunther Uhlmann
    • 2
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations