Bodrenko I.I. On analog of Darboux surfaces in many-dimensional euclidean spaces

Bodrеnko Irina Ivanovna

Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Fundamental Informatics and Optimal Control Volgograd State University
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Prospekt Universitetskij, 100, 400062 Volgograd, Russian Federation


Abstract. The Darboux tensor, symmetric covariant three-valent tensor Θ, was determined on two dimensional surfaces with nonzero Gaussian curvature K≠ 0 in Euclidean space E3. The term Θ ≡ 0 is the characteristic condition of Daroux surfaces in E3.

    The symmetric covariant three-valent tensor Θ(n) is determined on hypersurfaces Fn (n ≥ 2) with nonzero Gaussian curvature K≠ 0 in Euclidean space En+1. If n = 2 then tensor Θ(n) is coincided with the Darboux tensor Θ: Θ(2) ≡ Θ.

    Let D(n) be a set of hypersurfaces Fn (n ≥ 2) with nonzero Gaussian curvature K≠ 0 in Euclidean spaces En+1, on which the following condition holds Θ(n) ≡ 0. The set D(2) becomes exhausted by Daroux surfaces in E3. The properties of hypersurfaces FnEn+1 from the set D(n) for n ≥ 2 are studied in this article.

    The necessary and sufficient conditions, for which hypersurface Fn with nonzero Gaussian curvature K≠0 in Euclidean space En+1 belongs to the set D(n) (n ≥ 2), are derived. It was proved that hypersurface FnEn+1 with nonzero Gaussian curvature K≠ 0 belongs to the set D(n) (n ≥ 2) if and only if there exist coordinates of curvature (u1, . . . , un), in neighborhood O(x) ⊂ Fn of every point xFn, such that the following conditions hold:

where k1, . . ., kn  are the principal curvatures Fn, K=k1k2 . . . kn is Gaussian curvature of Fn, , are certain functions.

    It was proved that every cyclic recurrent hypersurface Fn En+1 with nonzero Gaussian curvature K ≠ 0 belongs to the set D(n) (n ≥ 2)
    The characteristic property of hypersphere Sn ⊂ En+1 was derived. It was proved that connected complete hypersurface Fn of constant positive Gaussian curvature K = const > 0 in Euclidean space En+1, belonging to the set D(n) (n ≥ 2), is sphere Sn ⊂ En+1.

Key words: Darboux tensor, Darboux surface, Gaussian curvature, second fundamental form, hypersurface, many-dimensional Euclidean space.

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On analog of Darboux surfaces in many-dimensional euclidean spaces by Bodrenko I.I. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №1 (18) 2013 pp. 24-30

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