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

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 Mwith Gaussian curvature of constant signs has recurrent Riemannian — Chrictoffel curvature tensor R. The following equality holds: ∇R = d ln |K| ⊗ R, where 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 F2E4, 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 F2E4 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.

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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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