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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

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Analytic embedding of pseudo-Helmholtz geometry

Kyrov Vladimir A., Gorno-Altaisk State University

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.

  1. Mikhailichenko G. G. The Mathematical Basics and Results of the Theory of Physical Structures. Gorno-Altaisk, Publishing house of Gorno-Altaisk State University, 2016. 267 p. Avaiable at: https://arxiv.org/pdf/1602.02795 (accessed 20 October 2020).
  2. Mikhailichenko G. G. Group and phenomenological symmetries in geometry. Siberian Mathematical Journal, 1984, vol. 25, iss. 5, pp. 764–774. https://doi.org/10.1007/ BF00968690
  3. Lev V. Kh. Three-dimensional geometries in the theory of physical structures. Computing Systems, 1988, no. 125, pp. 90–103 (in Russian).
  4. Kyrov V. A. Analytic embedding of some two-dimensional geometries of maximum mobility. Siberian Electronic Mathematical Reports, 2018, vol. 16, pp. 916–937 (in Russian). https://doi.org/10.33048/semi.2019.16.061
  5. Ovsyannikov L. V. Group Analysis of Differential Equation. New York, Academic Press, 1982. 400 p. (Russ. ed.: Moscow, Nauka, 1978. 400 p.).
  6. Fichtenholz G. M. Kurs differentsial’nogo i integral’nogo ischisleniya [A Course of Differential and Integral Calculus]. Moscow, Fizmatlit, 1963. Vol. 2. 524 p. (in Russian).
  7. Schwartz L. Analyse mathematique. T. I. Paris, Hermann, 1967. 824 p. (in French).
  8. Dyakonov V. Maple 10/11/12/13/14 v matematicheskikh raschetakh [Maple 10/11/12/13/14 in Mathematical Calculations]. Moscow, DMS Press, 2014. 640 p. (in Russian).