Boundary and lens rigidity, tensor tomography and analytic microlocal analysis
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.
KeywordsSymmetric Tensor Stability Estimate Conjugate Point Compact Riemannian Manifold Boundary Rigidity
Unable to display preview. Download preview PDF.
- 1.I. Alexandrova, Semi-Classical wavefront set and Fourier integral operators, to appear in Can. J. Math. Google Scholar
- 2._____, Structure of the Semi-Classical Amplitude for General Scattering Relations, Comm. PDE 30(2005), 1505–1535.Google Scholar
- 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
- 7.D. Burago and S. Ivanov, Boundary rigidity and filling volume minimality of metrics close to a flat one, manuscript, 2005.Google Scholar
- 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.G. Herglotz, Uber die Elastizitaet der Erde bei Beruecksichtigung ihrer variablen Dichte, Zeitschr. fur Math. Phys., 52(1905), 275–299.Google Scholar
- 16.C. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón Problem with partial data, to appear in Ann. Math. Google Scholar
- 25.V. Sharafutdinov, Integral geometry of tensor fields, VSP, Utrech, the Netherlands, 1994.Google Scholar
- 26.V. Sharafutdinov, M. Skokan, and G. Uhlmann, Regularity of ghosts in tensor tomography, Journal of Geometric Analysis textbf15(2005), 517–560.Google Scholar
- 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._____, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, in progress.Google Scholar
- 38.E. Wiechert E and K. Zoeppritz, Uber Erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss, Goettingen 4(1907), 415–549.Google Scholar