References
A. P. Kopylov, “Boundary values of mappings close to an isometric mapping,” Sib. mat. Zh.,25, No. 3, 120–131 (1984).
A. V. Pogorelov, Extrinsic Geometry of Convex Surfaces, Amer. Math. Soc., Providence (1973).
V. A. Aleksandrov, “Isometry of domains inR n and relative isometry of their boundaries. II,” Sib. Mat. Zh.,26, No. 6, 3–8 (1985).
V. A. Aleksandrov, “Isometry of domains inR n and relative isometry of their boundaries. I,” Sib. Mat. Zh.,25, No. 3, 3–13 (1984).
V. A. Aleksandrov, “Uniqueness of domains with non-Jordan boundaries,” Sib. Mat. Zh.,30, No. 1, 3–12 (1989).
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).
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.
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).
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).
I. Herbrurt and M. Moszynska, “On intrinsic embeddings,” Glas. Mat.,22 (42), 421–427 (1987).
M. Moszynska, “On rigid subsets of some manifolds,” Colloq. Math., LVII, fasc. 2, 247–254 (1989).
K. Rudnik, “Conserning the rigidity problem for subsets ofE 2,” Bull. Pol. Acad. Sci. Math.,37, No. 1–6, 251–254 (1989).
G. Herglots, “Über die Blastisitat der Erde bei Borucksichtigung ihrer variablen Dichte,” Z. fur Math. und Phys.,52, No. 3, 275–299 (1905).
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).
M. L. Gerver and V. M. Markushevich, “On the characteristic properties of seismic hodographs,” Dokl. Akad. Nauk SSSR,175, No. 2, 334–337 (1967).
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).
Yu. E. Anikonov, “On a problem of defining a Riemannian metric,” Dokl. Akad. Nauk SSSR,204, No. 6, 1287–1288 (1972).
Yu. E. Anikonov, “Definition of a metric of a Liouville surface,” Sib. Mat. Zh.,14, No. 6, 1338–1340 (1873).
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).
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).
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).
V. G. Romanov, “On some classes of uniqueness for the solution of integral geometry problems,” Mat. Zametki,16, No. 4, 657–668 (1974).
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).
V. G. Romanov, “Integral geometry on geodesics of an isotropic Riemannian metric,” Dokl. Akad. Nauk SSSR,241, No. 2, 290–293 (1978).
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).
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).
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).
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).
R. Richel, “Sur la rigidite imposse par la longueur des geodesiques,” Invent. Math.,65, fasc. 1, 71–83 (1981).
M. Gromov, “Filling Riemannian manifolds,” J. Different. Geom.,18, No. 1, 1–147 (1983).
C. B. Croke, “Rigidity for surfaces of non-positive curvature,” Comment. Math. Helv.,65, No. 1, 150–169 (1990).
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).
N. H. Kuiper, “Review 6641, a−d,” Math. Reviews,37, No. 5, 1126–1127 (1967).
Additional information
Novosibirsk. Translated fromSibirskiį Matematicheskiį Zhurnal, Vol. 33, No. 4, pp. 206–211, July–August, 1992.
Translated by the author
Rights and permissions
About this article
Cite this article
Aleksandrov, V.A. On interrelation between the problem of unique determination of a domain in RN and a problem of recovery of a locally euclidean metric. Sib Math J 33, 732–736 (1992). https://doi.org/10.1007/BF00971139
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00971139