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On function-degenerate immersions of a Lobachevsky plane into E4

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Abstract

A surface F2 in E4 is called function-degenerate if parameters α, β, a, and b of the ellipse of normal curvature (α and β are the coordinates of its center in the normal plane and a and b are the values of the semiaxes) are some functions of a function t(P) of point P of the surface. This class of surfaces is a generalization of the class of surfaces that admit motion with respect to themselves along some family of lines. The following theorem is proved: There does not exist a regular, class C6, isometric immersion of the Lobachevsky plane L2 into E4, with zero Gaussian torsion, in the form of a function-degenerate surface. This theorem includes the Hilbert theorem on surfaces of negative curvature in E3.

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Literature cited

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    Yu. A. Aminov, “On immersions of n-dimensional Lobachevsky space into 2n-dimensional Euclidean space with n fields of principal directions”, Ukr. Geom. Sb., No. 28, 3–8 (1985).

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    S. B. Kadomtsev, “Investigation of uniqueness problems of two-dimensional surfaces in Euclidean spaces”, Probl. Geometrii,8, 243–256 (1977).

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    E. P. Rozendorn, “On the problem of immersion of two-dimensional Riemannian metrics in four-dimensional Euclidean space”, Vestn. Mosk. Univ. Ser. I Mat. Mekh., No. 2, 47–50 (1979).

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    Yu. A. Aminov, “Isometric immersions, with flat normal connections, of domains on n-dimensional Lobachevsky space in Euclidean spaces. A model of a gauge field”, Mat. Sb.,137. No. 3, 275–298 (1988).

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Additional information

Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 8–18, 1990.

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Aminov, Y.A. On function-degenerate immersions of a Lobachevsky plane into E4 . J Math Sci 53, 467–474 (1991). https://doi.org/10.1007/BF01109644

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Keywords

  • Normal Curvature
  • Negative Curvature
  • Isometric Immersion
  • Normal Plane
  • Lobachevsky Plane