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

Details

Hits: 1334

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
This email address is being protected from spambots. You need JavaScript enabled to view it.
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

Attachments:

I.I. Bodrenko URL: https://mp.jvolsu.com/index.php/en/component/attachments/download/102

1000 Downloads