Stability estimates for the X-ray transform of tensor fields and boundary rigidity

Duke Mathematical Journal

Volumn 123, Issue 3, 2004, Pages 445-467

  Stefanov, Plamen ^a   Uhlmann, Gunther ^b  

^a PURDUE UNIVERSITY   (United States)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 3142762630     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-04-12332-2     Document Type: Article

References (26)

1. [AR] YU. ANIKONOV and V. ROMANOV, On uniqueness of determination of a form of first degree by its integrals along geodesies, J. Inverse Ill-Posed Probl. 5 (1997), 487-480. MR 1623603
2. [BG] I. N. BERNSTEIN and M. L. GERVER, "Conditions on distinguishability of metrics by hodographs (in Russian)" in Methods and Algorithms of Interpretation of Seismological Information, Computerized Seismology 13, Nauka, Moscow, 1980, 50-73.
3. [C] E. CHAPPA, On the characterization of the kernel of the geodesic X-ray transform for tensor field, to appear in Trans. Amer. Math. Soc.
4. [Cr] K. C. CREAGER, Anisotropy of the inner core from differential travel times of the phases PKP and PKIPK, Nature 356 (1992), 309-314.
5. [Cro1] C. B. CROKE, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), 150-169. MR 1036134
6. [Cro2] _, Rigidity and the distance between boundary points, J. Differential Geom. 33 (1991), 445-464. MR 1094465
7. [CDS] C. B. CROKE, N. S. DAIRBEKOV, and V. A. SHARAFUTDINOV, Local boundary rigidity of a compact Riemannian manifold with curvature bounded above, Trans. Amer. Math. Soc. 352 (2000), 3937-3956. MR 1694283
8. [E] G. ESKIN, Inverse scattering problem in anisotropic media, Comm. Math. Phys. 199 (1998), 471-491. MR 1666879
9. [G] M. GROMOV, Filling Riemannian manifolds, J. Differential Geometry 18 (1983), 1-147. MR 0697984
10. [GS] V. GUILLEMIN and S. STERNBERG, Geometric Asymptotics, Math. Surveys 14, Amer. Math. Soc., Providence, 1977. MR 0516965
11. [H] G. HERGLOTZ, Über die Elastizität der Erde bei Berücksichtigung ihrer variablen Dichte, Zeitschr. fur Math. Phys. 52 (1905), 275-299.
12. [LSU] M. LASSAS, V. SHARAFUTDINOV, and G. UHLMANN, Semiglobal boundary rigidity for Riemannian metrics, Math. Ann. 325 (2003), 767-793. MR 1974568
13. [M] R. MICHEL, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65 (1981), 71-83. MR 0636880
14. [Mu1] R. G. MUHOMETOV, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (in Russian), Dokl. Akad. Nauk SSSR 232, no. 1 (1977), 32-35; English translation in Soviet Math. Dokl. 18, no. 1 (1977), 27-31. MR 0431074
15. [Mu1] R. G. MUHOMETOV, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (in Russian), Dokl. Akad. Nauk SSSR 232, no. 1 (1977), 32-35; English translation in Soviet Math. Dokl. 18, no. 1 (1977), 27-31. MR 0431074
16. [Mu2] _, On a problem of reconstructing Riemannian metrics, Siberian Math. J. 22, no. 3 (1981), 420-433. MR 0621466
17. [MR] R. G. MUHOMETOV and V. G. ROMANOV, On the problem of finding an Isotropic Riemannian metric in an n-dimensional space (in Russian), Dokl. Akad. Nauk SSSR 243, no. 1 (1978), 41-44. MR 0511273
18. [O] J.-P. OTAL, Sur les longueur des géodésiques d'une métrique à coubure négative dans le disque, Comment. Math. Helv. 65 (1990), 334-347. MR 1057248
19. [PS] L. N. PESTOV and V. A. SHARAFUTDINOV, Integral geometry of tensor fields on a manifold of negative curvature (in Russian), Sibirsk. Mat. Zh. 29, no. 3 (1988), 114-130; English translation in Siberian Math. J. 29, no. 3 (1988), 427-441. MR 0953028
20. [PS] L. N. PESTOV and V. A. SHARAFUTDINOV, Integral geometry of tensor fields on a manifold of negative curvature (in Russian), Sibirsk. Mat. Zh. 29, no. 3 (1988), 114-130; English translation in Siberian Math. J. 29, no. 3 (1988), 427-441. MR 0953028
21. [S1] V. A. SHARAFUTDINOV, Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser., VSP, Utrecht, Netherlands, 1994. MR 1374572
22. [S2] _, Finiteness theorem for the ray transform on a Riemannian manifold. Inverse Problems 11 (1995), 1039-1050. MR 1353801
23. [SU] V. SHARAFUTDINOV and G. UHLMANN, On deformation boundary rigidity and spectral rigidity for Riemannian surfaces with no focal points, J. Differential Geom. 56 (2000), 93 -110. MR 1863022
24. [StU] P. STEFANOV and G. UHLMANN, Rigidity for metrics with the same lengths of geodesies. Math. Res. Lett. 5 (1998), 83-96. MR 1618347
25. [T] M. E. TAYLOR, Pseudodifferential Operators, Princeton Math. Ser. 34, Princeton Univ. Press, Princeton, 1981. MR 0618463
26. [WZ] E. WIECHERT and K. ZOEPPRITZ, Über Erdbebenwellen, Koenigl. Nachr. Gesellsch. Wiss. Goettingen 4 (1907), 415-549.