Advertisement

Siberian Mathematical Journal

, Volume 33, Issue 4, pp 732–736 | Cite as

On interrelation between the problem of unique determination of a domain in R N and a problem of recovery of a locally euclidean metric

  • V. A. Aleksandrov
Article
  • 12 Downloads

Keywords

Unique Determination 
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.
    A. P. Kopylov, “Boundary values of mappings close to an isometric mapping,” Sib. mat. Zh.,25, No. 3, 120–131 (1984).Google Scholar
  2. 2.
    A. V. Pogorelov, Extrinsic Geometry of Convex Surfaces, Amer. Math. Soc., Providence (1973).Google Scholar
  3. 3.
    V. A. Aleksandrov, “Isometry of domains inR n and relative isometry of their boundaries. II,” Sib. Mat. Zh.,26, No. 6, 3–8 (1985).Google Scholar
  4. 4.
    V. A. Aleksandrov, “Isometry of domains inR n and relative isometry of their boundaries. I,” Sib. Mat. Zh.,25, No. 3, 3–13 (1984).Google Scholar
  5. 5.
    V. A. Aleksandrov, “Uniqueness of domains with non-Jordan boundaries,” Sib. Mat. Zh.,30, No. 1, 3–12 (1989).Google Scholar
  6. 6.
    V. A. Aleksandrov, “Unique determination of domains by the relative metrics of their boundaries,” Tr. In-ta Mat. AN SSSR, Sib. otd-nie,7, Issledovaniya po Geometrii i Matematicheskomu Analizu, 5–18 (1987).Google Scholar
  7. 7.
    D. A. Trotsenko, “Unique determination of bounded domains by the metric of boundary, induced by the metric of domain” in: Abstracts: Proc. All-Union Conf. of Geometry in the Large [in Russian], Novosibirsk (1987), p. 122.Google Scholar
  8. 8.
    A. V. Kuz'minykh, “Isometry of domains whose boundaries are isometric in the relative metrics,” Sib. mat. Mat. Zh.,26, No. 3, 91–99 (1985).Google Scholar
  9. 9.
    V. A. Aleksandrov, “Estimation of the deformation of a strictly convex domain depending on the variation of relative metric of its boundary,” Sib. Mat. Zh.,31, No. 5, 3–9 (1990).Google Scholar
  10. 10.
    I. Herbrurt and M. Moszynska, “On intrinsic embeddings,” Glas. Mat.,22 (42), 421–427 (1987).Google Scholar
  11. 11.
    M. Moszynska, “On rigid subsets of some manifolds,” Colloq. Math., LVII, fasc. 2, 247–254 (1989).Google Scholar
  12. 12.
    K. Rudnik, “Conserning the rigidity problem for subsets ofE 2,” Bull. Pol. Acad. Sci. Math.,37, No. 1–6, 251–254 (1989).Google Scholar
  13. 13.
    G. Herglots, “Über die Blastisitat der Erde bei Borucksichtigung ihrer variablen Dichte,” Z. fur Math. und Phys.,52, No. 3, 275–299 (1905).Google Scholar
  14. 14.
    M. L. Gerver and V. M. Markushevich, “Investigation of non-uniqueness in determination of a seismic wave velocity from the hodograph,” Dokl. Akad. Nauk SSSR,163, No. 6, 1377–1380 (1965).Google Scholar
  15. 15.
    M. L. Gerver and V. M. Markushevich, “On the characteristic properties of seismic hodographs,” Dokl. Akad. Nauk SSSR,175, No. 2, 334–337 (1967).Google Scholar
  16. 16.
    A. V. Belonosova and A. S. Alekseev, “On an inverse kinematic problem of seismics for two-dimensional continuously-inhomogeneous media,” in: Methods and Algorithms of Interrelation of Geophysical Data [in Russian], Nauka, Moscow (1967).Google Scholar
  17. 17.
    Yu. E. Anikonov, “On a problem of defining a Riemannian metric,” Dokl. Akad. Nauk SSSR,204, No. 6, 1287–1288 (1972).Google Scholar
  18. 18.
    Yu. E. Anikonov, “Definition of a metric of a Liouville surface,” Sib. Mat. Zh.,14, No. 6, 1338–1340 (1873).Google Scholar
  19. 19.
    M. M. Lavrent'ev and A. L. Bukhgeim, “On a class of problems in integral geometry,” Dokl. Akad. Nauk SSSR,211, No. 1, 38–39 (1973).Google Scholar
  20. 20.
    Yu. E. Anikonov, “On the problem of determining the Riemannian metric\(ds^2 = \lambda ^2 (x)|dx|^2\),” Mat. Zametki,16, No. 4, 611–617 (1974).Google Scholar
  21. 21.
    V. G. Romanov, “On the uniqueness of the defining of an isotropic Riemannian metric inside a domain in terms of distances between points of the boundary,” Dokl. Akad. Nauk SSSR,218, No. 2, 295–297 (1974).Google Scholar
  22. 22.
    V. G. Romanov, “On some classes of uniqueness for the solution of integral geometry problems,” Mat. Zametki,16, No. 4, 657–668 (1974).Google Scholar
  23. 23.
    R. G. Mukhometov, “The problem of recovery of a two-dimensional Riemannian metric and integral geometry,” Dokl. Akad. Nauk SSSR,232, No. 1, 32–35 (1977).Google Scholar
  24. 24.
    V. G. Romanov, “Integral geometry on geodesics of an isotropic Riemannian metric,” Dokl. Akad. Nauk SSSR,241, No. 2, 290–293 (1978).Google Scholar
  25. 25.
    R. G. Mukhometov and V. G. Romanov, “On the problem of finding an isotropic Riemannian metric in ann-dimensional space,” Dokl. Akad. Nauk SSSR,243, No. 1, 41–44 (1978).Google Scholar
  26. 26.
    M. L. Gerver and N. S. Nadirashvili, “A condition for isometry of Riemannian metrics in the disc,” Dokl. Akad. Nauk SSSR,275, No. 2, 289–293 (1984).Google Scholar
  27. 27.
    V. A. Sharafutdinov, “Integral geometry of a quadratic differential form for two-dimensional metrics close to the Euclidean metric,” Dokl. Akad. Nauk SSSR,300, No. 3, 551–554 (1988).Google Scholar
  28. 28.
    V. A. Sharafutdinov, “Integral geometry of a tensor field along geodesics of metric close to the Euclidean metric,” Dokl. Akad. Nauk SSSR,304, No. 6, 1308–1311 (1988).Google Scholar
  29. 29.
    R. Richel, “Sur la rigidite imposse par la longueur des geodesiques,” Invent. Math.,65, fasc. 1, 71–83 (1981).Google Scholar
  30. 30.
    M. Gromov, “Filling Riemannian manifolds,” J. Different. Geom.,18, No. 1, 1–147 (1983).Google Scholar
  31. 31.
    C. B. Croke, “Rigidity for surfaces of non-positive curvature,” Comment. Math. Helv.,65, No. 1, 150–169 (1990).Google Scholar
  32. 32.
    L. N. Pestov and V. A. Sharafutdinov, “Integral geometry of tensor fields on a manifold of negative curvature,” Dokl. Akad. Nauk SSSR,295, No. 6, 1318–1320 (1987).Google Scholar
  33. 33.
    N. H. Kuiper, “Review 6641, a−d,” Math. Reviews,37, No. 5, 1126–1127 (1967).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • V. A. Aleksandrov

There are no affiliations available

Personalised recommendations