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## Archive of Issues

Russia Gorno-Altaisk
Year
2019
Volume
29
Issue
4
Pages
532-547
 Section Mathematics Title Analytical embedding of three-dimensional Helmholtz-type geometries Author(-s) Kyrov V.A.a Affiliations Gorno-Altaisk State Universitya Abstract For modern geometry, the study of maximum mobility geometries is important. The maximum mobility for $n$-dimensional geometry given by the function $f$ of a pair of points means the existence of an $n(n+1)/2$-dimensional transformation group, which leaves this function invariant. Many geometries of maximum mobility are known (Euclidean, symplectic, Lobachevsky, etc.), but there is no complete classification of such geometries. In this article, the method of embedding solves one of these classification problems. The essence of this method is as follows: from the function of a pair of points $g$ of three-dimensional geometry, we find all non-degenerate functions $f$ of a pair of points of four-dimensional geometries that are invariants of the Lie group of transformations of dimension 10. In this article, $g$ are non-degenerate functions of a pair of points of two Helmholtz three-dimensional geometries: $$g = 2\ln(x_i-x_j) + \dfrac{y_i-y_j}{x_i-x_j} + 2z_i + 2z_j,$$ $$\ln [(x_i-x_j)^2 + (y_i-y_j)^2] + 2\gamma\,\text{arctg}\dfrac{y_i-y_j}{x_i-x_j} + 2z_i + 2z_j.$$ These geometries are locally maximally mobile, that is, their groups of motions are six-dimensional. The problem solved in this work is reduced to solving special functional equations by analytical methods, the solutions of which are sought in the form of Taylor series. For searching various options, the math software package Maple 15 is used. As a result, only degenerate functions of a pair of points are obtained. Keywords functional equation, function of a pair of points, group of motions, geometry of maximum mobility, Helmholtz geometry UDC 517.912, 514.1 MSC 39A05, 39B05 DOI 10.20537/vm190405 Received 4 July 2019 Language Russian Citation Kyrov V.A. 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