PROPERTIES OF THE SPHERICAL IMAGE OF A SPATIAL STRIP IN E⁴ 

Sharmin Valentin Gennad'evich, Candidate of physical and mathematical sciences, associate professor, sub-department of algebra and mathematical logic, Tyumen State University (6 Volodarskogo street, Tyumen, Russia), sharmin@utmn.ru
Sharmin Dmitriy Valentinovich, Candidate of pedagogical sciences, associate professor, sub-department of mathematics and informatics, Tyumen State University (6 Volodarskogo street, Tyumen, Russia), dsharmin@mail.ru

514.752

10.21685/2072-3040-2018-1-3

Background. The study of the properties of surfaces in various spaces is one of the main problems of differential geometry. Surfaces in Euclidean space, whose codimension is greater than one, are characterized by some new properties that do not have hypersurfaces in this space. In particular, two-dimensional surfaces in fourdimensional Euclidean space have torsion coefficients. This article is devoted to the study of the properties of a spherical image of a two-dimensional surface with a system normals without torsion in four-dimensional Euclidean space.
Materials and methods. The methods of differential geometry developed by E. Cartan, K. Sh. Ramazanova, and A. I. Firsov to study surfaces, whose codimension is greater than one.
Results. We have proved some properties of the spherical image of a twodimensional surface with a system of normals without torsion, and also we have obtained sufficient conditions that this image is a three-dimensional surface.
Conclusions. We have investigated the structure of the spherical image of a twodimensional surface with a system of normals without torsion, under certain additional conditions.

Euclidean space, two-dimensional surface, spherical mapping, Gaussian curvature, coefficients of torsion of the surface

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