Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Kyrov V. A. Analytic embedding of pseudo-Helmholtz geometry. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 3, pp. 294-304. DOI: 10.18500/1816-9791-2021-21-3-294-304, EDN: EKKKGS

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2021
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Russian
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Article
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517.912+514.1
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EKKKGS

Analytic embedding of pseudo-Helmholtz geometry

Autors: 
Kyrov Vladimir A., Gorno-Altaisk State University
Abstract: 

For modern geometry, the study of maximal mobility geometries is of great importance. Some of these geometries are well studied (Euclidean, pseudo-Euclidean, symplectic, spherical, Lobachevsky, etc.), and others are poorly understood (Helmholtz, pseudo-Helmholtz, etc.). There is no complete classification of geometries for maximum mobility. In this paper part of this large classification problem is solved. The solution is sought by the embedding method, the essence of which is to find the functions of a pair of $f = \chi(g,w_i,w_j)$, specifies $(n+1)$-dimensional geometries of maximum mobility, using the well-known function of a pair of $g$ $n$-dimensional geometries of maximum mobility. In this paper, $g$ is either a function of a pair of points of two-dimensional pseudo-Helmholtz geometry $g = \beta\ln|y_i-y_j|  +\varepsilon\ln|x_i-x_j|,$ or the function of a pair of points of three-dimensional pseudo-Helmholtz geometry $g = \beta\ln|y_i-y_j|  +\varepsilon\ln|x_i-x_j| + 2z_i + 2z_j$. Both of these geometries are maximum mobility geometries. As a result of embedding a two-dimensional pseudo-Helmholtz geometry, we obtain a three-dimensional pseudo-Helmholtz geometry, but as a result of embedding a three-dimensional pseudo-Helmholtz geometry, geometries of maximum mobility are not obtained. Solving the embedding problem is reduced to solving special functional equations in the class of analytic functions.

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Received: 
21.12.2020
Accepted: 
26.04.2021
Published: 
31.08.2021