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Small parameter in the theory of isometric imbeddings for two-dimensional Riemannian manifolds into euclidean spaces

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Literature Cited

  1. I. N. Vekua,Generalized Analytic Functions [in Russian], Nauka, Moscow (1962).

    Google Scholar 

  2. D. Hilbert,Grundlagen der Geometrie. Applikation V. Über Flächen von konstanter Gauszsher Krümung, Leipzig-Berlin (1909).

  3. M. L. Gromov and V. A. Rochlin, “Imbeddings and immersions in Riemannian geometry,”Uspekhi Mat. Nauk,25, No. 5, 3–62 (1970).

    Google Scholar 

  4. N. V. Efimov and E. G. Poznjak, “Generalization of the Hilbert theorem on surfaces with constant negative curvature,”Dokl. Akad. Nauk SSSR,137, No. 3, 509–512 (1962).

    MathSciNet  Google Scholar 

  5. V. F. Kagan,Foundations of Surface Theory, Vol. 2, Gostechizdat, Moscow (1947).

    Google Scholar 

  6. J. Kajdasov, “On the regular isometric imbedding of the widening strip on a Lobachevsky plane intoE 3,”Issled. po Teorii Poverkhnostei v Riman. Prostranstvakh, LGPI, Leningrad (1984), pp. 119–129.

    Google Scholar 

  7. J. Kajdasov and E. V. Shikin, “On the isometric imbedding of the convex domain on a Lobachevsky plane containing two oricircles intoE 3,”Mat. Zametki,39, No. 4, 612–617 (1986).

    MathSciNet  Google Scholar 

  8. J. Kaidasov and E. V. Shikin, “On the isometric imbedding of the convex domain on a Lobachevsky plane containing an oricircle intoE 3,”Vestn. MGU, Ser. Mat. Mekh., No. 5, 79–81 (1986).

    Google Scholar 

  9. J. Kaidasov and E. V. Shikin, “On the isometric imbedding of widening strips on manifolds of theL-type intoE 3,”Mat. Zametki,42, No. 6, 842–853 (1987).

    MathSciNet  Google Scholar 

  10. R. Courant,Partial Differential Equations, New York-London (1962).

  11. A. V. Pogorelov,Exterior Geometry of Convex Surfaces [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  12. A. V. Pogorelov, “An example of a two-dimensional metric with no local realization inE 3,”Dokl. Akad. Nauk SSSR,198, No. 1, 42–43 (1971).

    MATH  MathSciNet  Google Scholar 

  13. E. G. Poznyak, “Examples of the regular metrics on a sphere and in a disk with no realization in the class of doubly differentiable surfaces,”Vestn. MGU, Ser. Mat. Mekh., No. 2, 3–5 (1960).

    MATH  Google Scholar 

  14. E. G. Poznyak, “On regular realization in the large of the two-dimentional metrics with negative curvature,”Dokl. Akad. Nauk SSSR,170, No. 4, 786–789 (1966).

    MATH  MathSciNet  Google Scholar 

  15. E. G. Poznyak, “On regular realization in the large of the two-dimensional metrics with negative curvature,”Ukr. Geom. Sb., No. 3, 78–92 (1966).

    MATH  Google Scholar 

  16. E. G. Poznyak, “Realization in the large of the two-dimensional analytic metrics with alternate curvature,”Ukr. Geom. Sb., No. 7, 89–97 (1970).

    MATH  Google Scholar 

  17. E. G. Poznyak, “Isometric imbeddings of the two-dimensional metrics into Euclidean spaces,”Uspekhi Mat. Nauk,28, No. 41, 47–76 (1973).

    MATH  Google Scholar 

  18. E. G. Poznyak, “Geometric investigations connected with the equationz xy =sinz,” In:Problemy Geometrii, Vol. 8,Itogi Nauki i Tekhn., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1977), pp. 225–241.

    Google Scholar 

  19. E. G. Poznyak, “Some new results on the isometric imbeddings of the Lobachevski plane parts intoE 3,”Vses. Nauch. Konf. po Neevklid. Geometrii “150 Let Geometrii Lobachevskogo,” Kazan’, 1976, Plenarnye Dokl., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1977), pp. 73–78.

    Google Scholar 

  20. E. G. Poznyak, “Isometric imbedding of several non-compact parts of the Lobachevski plane,”Mat. Sb.,102, No. 1, 3–12 (1977).

    MATH  MathSciNet  Google Scholar 

  21. B. L. Rozhdestvenskii, “System of quasilinear equations of surface theory,”Dokl. Akad. Nauk SSSR,143, No. 1, 50–52 (1962).

    MATH  MathSciNet  Google Scholar 

  22. D. V. Tunitskii, “On the regular isometric imbedding of the unbounded domains of negative curvature intoE 3,”Mat. Sb.,134, No. 1, 119–134 (1987).

    MATH  MathSciNet  Google Scholar 

  23. D. V. Tunitskii, “On the regular isometric imbedding in the large of the two-dimensional metrics with nonpositive curvature,”Mat. Sb.,183, No. 7, 65–80 (1992).

    MATH  Google Scholar 

  24. P. Tschebyscheff, “Sur la coupe des vetements,”Assoc. Franc., Oeuvres II (1878).

  25. E. V. Shikin, “On the regular realization in the large of the two-dimensional metrics of classC 2 with negative curvature of classC 1 inE 3,”Dokl. Akad. Nauk SSSR,188, No. 5, 1014–1016 (1969).

    MATH  MathSciNet  Google Scholar 

  26. E. V. Shikin, “On the regular imbedding in the large ofC 4-smooth metrics with negative curvature in ℝ3,”Mat. Zametki,14, No. 2, 261–266 (1973).

    MathSciNet  Google Scholar 

  27. E. V. Shikin, “On the isometric imbedding in the large of some metrics with nonpositive curvature in ℝ3,”Dokl. Akad. Nauk SSSR,215, No. 1, 61–65 (1974).

    MathSciNet  Google Scholar 

  28. E. V. Shikin, “Isometric imbeddings of non-compact domains with non-positive curvature inE 3,” In:Problemy Geometrii, Vol. 7,Itogi Nauki i Tekhn., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1975), pp. 249–266.

    Google Scholar 

  29. E. V. Shikin, “On the existence of a solution for the Peterson-Codazzi and Gauss system,”Mat. Zametki,17, No. 5, 765–781 (1975).

    MATH  MathSciNet  Google Scholar 

  30. E. V. Shikin, “Isometric imbeddings of non-compact domains with non-positive curvature inE 3,”Mat. Zametki,25, No. 5, 785–797 (1979).

    MATH  MathSciNet  Google Scholar 

  31. E. V. Shikin, “On the isometric imbedding of two-dimensional metrics with negatice curvature by the method of Darboux,”Mat. Zametki,27, No. 5, 779–794 (1980).

    MATH  MathSciNet  Google Scholar 

  32. E. V. Shikin, “On the isometric imbedding equations of the two-dimensional manifolds with non-positive curvature in three-dimensional Euclidean space,”Mat. Zametki,31, No. 4, 601–612 (1982).

    MATH  MathSciNet  Google Scholar 

  33. M. H. Amsler, “Des surfaces à courbure negative constante dans l’espace à trois dimensions et de leurs singularités,”Math. Ann.,130, No. 3, 234–256 (1955).

    MATH  MathSciNet  Google Scholar 

  34. A. V. Bäcklund, “Zur Theorie der Flächen transformationen,”Math. Ann.,19 (1882).

  35. E. Beltrami, “Sulla superficie di rotazione che serve di tipo alle superficie pseudosferiche,”Giorno Mat.,10 (1872).

  36. L. Bianchi,Lezioni di Geometria Differenziale, Bologna (1927).

  37. G. Darboux,Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calculus Infinitésimal, P. 3, Gautier-Villars, Paris (1887).

    Google Scholar 

  38. K. O. Friedrichs, “Symmetric positive linear differential equations,”Commun. Pure Appl. Math.,11, No. 3, 333–418 (1956).

    MathSciNet  Google Scholar 

  39. R. S. Hamilton, “The inverse function theorem of Nash and Moser,”Bull. Amer. Math. Soc.,7, No. 1, 165–222 (1982).

    MathSciNet  Google Scholar 

  40. E. Levi, “Sur l’application des équations integrales an problèm de Riemann,”Nachr. Königl. Gesell. Wiss., Mat, Göttingen, 249–252 (1908).

    Google Scholar 

  41. C. S. Lin, “The local isometric embedding in ℝ3 of two-dimensional Riemannian manifolds with non-negative curvature,”J. Diff. Geom.,21, 213–230 (1985).

    MATH  Google Scholar 

  42. C. S. Lin, “The local isometric embedding in ℝ3 of two-dimensional Riemannian manifolds with Gaussian curvature changing sign cleanly,”Commun. Pure Appl. Math.,39, No. 6, 867–887 (1986).

    MATH  Google Scholar 

  43. G. Nakamura, “Local isometric embedding of 2-dimensional Riemannian manifolds intoR 3 with nonpositive Gaussian curvature,”Proc. Jpn. Acad., Ser. A, No. 7, 211–212 (1985).

    Google Scholar 

  44. R. Sternwald, “Über Enneper’sche Flächen und Bäcklund’sche Transformation,”Abh. Bayer. Akad. Wiss.,40 (1936).

  45. Ch. Wissler, “Globale Tschebyscheff-Netze auf Riemannschen Mannigfaltigkeiten und Forsetzung von Flächen Konstanter Negativer Krürnung,”Comment Math. Helv.,47, No. 3, 348–372 (1972).

    MATH  MathSciNet  Google Scholar 

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Authors
  1. E. G. Poznyak
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  2. E. V. Shikin
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Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. 1993.

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Poznyak, E.G., Shikin, E.V. Small parameter in the theory of isometric imbeddings for two-dimensional Riemannian manifolds into euclidean spaces. J Math Sci 74, 1078–1116 (1995). https://doi.org/10.1007/BF02362833

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