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Absence of complete Lobachevskij planes in two classes of surfaces in \(E_ 4\) with \(K=-1\). (English. Russian original) Zbl 0717.53002

Math. Notes 48, No. 1, 666-671 (1990); translation from Mat. Zametki 48, No. 1, 68-74 (1990).
See the review in Zbl 0711.53044.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
Full Text: DOI

References:

[1] D. Hilbert, ?Über Flächen von konstanter Gausschen Krümmung,? Trans. Am. Math. Soc.,2, 87-99 (1901). · JFM 32.0608.01
[2] D. Hilbert, Foundations of Geometry [Russian translation], Ob?edin. Gos. Izdat., Moscow-Leningrad (1948).
[3] T. Klotts-Milnor, ?Efimov’s theorem on complete immersed surfaces of negative curvature,? Usp. Mat. Nauk,41, No. 5, 3-57 (1986). · Zbl 0612.53033
[4] É. G. Poznyak and E. V. Shikin, ?Surfaces of negative curvature,? in: Algebra, Topology, Geometry: Progress in Science and Technology [in Russian], Vol. 12, Vsesoyuz. Inst. Nauch. Tekh. Inf., Moscow (1974); pp. 171-207.
[5] É. R. Rozendorn, ?Realization of the metric ds2 + du2 + f2(u)dv2 in the five-dimensional Euclidean space,? Dokl. Akad. Nauk ArmSSR,30, No. 4, 197-199 (1960).
[6] D. Blanu?a, ?Über die Einbettung hyperbolischer Räume in euklidische Räume,? Monats. Math.,59, No. 3, 217-229 (1955). · Zbl 0067.14403 · doi:10.1007/BF01303796
[7] S. B. Kadomtsev, ?Impossibility of certain special isometric immersions of the Lobachevskii spaces,? Mat. Sb.,107, No. 2, 175-198 (1978).
[8] É. R. Rozendorn, ?On the problem of immersion of two-dimensional Euclidean space,? Vestn. Mosk. Gos. Univ., Ser. 1, Mat. Mekh., No. 2, 47-50 (1979). · Zbl 0406.53045
[9] Ü. Lumiste [Yu. Lumiste], ?Bäcklund’s theorem and transformation for surfaces V2 in En,? Acta Sci. Math.,50, 51-57 (1986). · Zbl 0614.53004
[10] Riemannian Geometry in Orthogonal Frame, Based on Cartan’s Lectures [Russian translation], Izd. Mosk. Gos. Univ., Moscow (1960).
[11] E. Cartan, Exterior Differential Systems and Their Geometrie Applications [Russian translation], Izd. Mosk. Gos. Univ., Moscow (1962).
[12] S. P. Finikov, The Method of the Cartan Exterior Forms in Differential Geometry [in Russian], Ob?edin. Gos. Izdat., Moscow-Leningrad (1948). · Zbl 0033.06004
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