Bodrenko I.I. A characteristic feature of the surfaces with constant gaussian torsion in

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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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Prospect Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. It is known that every two-dimensional Riemannian manifold M² with Gaussian curvature K of constant signs has recurrent Riemannian — Chrictoffel curvature tensor R. The following equality holds: ∇R = d ln |K| ⊗ R, where g is Riemannian metric M², ∇ is Riemannian connection.

     Let E⁴ be 4-dimensional Euclidean space with Cartesian coordinates (x¹, x², x³, x⁴), F² is two-dimensional surface in E⁴ given by vector equation

    The properties of surfaces F² with nonzero Gaussian torsion æ ≠ 0 in Euclidean space E⁴ are studied in this article.
    Let R^⊥ be normal curvature tensor of F² ⊂ E⁴, D is normal connection, is connection of van der Waerden — Bortolotti.

    Normal curvature tensor  R^⊥≠ 0 is called parallel if . Normal curvature tensor R^⊥ ≠ 0 is called recurrent (in connection ∇) if there exists 1-form ν on F² such that .
    The following statement is proved in this article. A surface F² with nonzero Gaussian torsion æ ≠ 0 in E⁴ has recurrent normal curvature tensor R^⊥:

    The characteristic feature of 2-dimensional surfaces F² with constant Gaussian torsion æ ≡ const ≠ 0 in 4-dimensional Euclidean space E⁴ was obtained in this article.
    It was proved that surface F² ⊂ E⁴ has constant Gaussian torsion æ ≡ const ≠ 0 if and only if normal curvature tensor R^⊥ ≠ 0 is parallel in connection of van der Waerden — Bortolotti.

Key words: Gaussian torsion, ellipse of normal curvature, normal curvature tensor, normal connection, connection of van der Waerden — Bortolotti.

A characteristic feature of the surfaces with constant gaussian torsion in E^4 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. №2 (19) 2013 pp. 13-17

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I.I. Bodrеnko URL: https://mp.jvolsu.com/index.php/en/component/attachments/download/147

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