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

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http://dx.doi.org/10.15688/jvolsu1.2013.1.2

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 Fⁿ ⊂ Eⁿ⁺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 Fⁿ⊂ Eⁿ⁺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 x ∈ Fⁿ, such that the following conditions hold:

where k₁, . . ., k_n  are the principal curvatures Fⁿ, K=k₁k₂ . . . k_n 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.

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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