Development and Isometry of Surfaces Galilean Space G3

Sultanov, B.M and Kurudirek, Abdullah and Ismoilov, Sh.Sh (2023) Development and Isometry of Surfaces Galilean Space G3. Mathematics and Statistics, 11 (6). pp. 965-972. ISSN 2332-2144

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Abstract

Currently, the study of the geometry of semi-Euclidean spaces is an urgent task of geometry. In the singular parts of pseudo-Euclidean spaces, a geometry associated with a degenerate metric appears. A special case of this geometry is the geometry of Galileo. The basic concepts of the geometry of Galilean space are given in the monograph by A. Artykbaev. Here the differential geometry "in the small" is studied, the first and second fundamental forms of surfaces and geometric characteristics of surfaces are determined. The derivational equations of surfaces, analogs of the Peterson-Codazzi and Gauss formulas are calculated. This paper studies the development and isometry of surfaces in Galilean space. Moreover, the isometry of surfaces in Galilean space is divided into three types: semi-isometry, isometry and completely isometry. This separation is due to the degeneracy of the Galilean space metric. The existence of a development of a surface projecting uniquely onto a plane in general position is proved, as well as the conditions for isometric and completely isometric surfaces of Galilean space. We present the conditions associated with the analog of the Christoffel symbol, providing isometries of the surfaces of Galilean space. An example of isometric, but not completely isometric surfaces in G3 is given. The concept of surface development for Galilean space is generalized. A development of the surface is obtained, which is uniquely projected onto the plane of the general position. In addition, the Gaussian curvature of the surface has been shown to be completely defined by Christoffel symbols.

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