The embedding of multidimensional special extensions of pseudo-Euclidean geometries

V. A. Kyrov

Čelâbinskij fiziko-matematičeskij žurnal, Tome 3 (2018) no. 4, pp. 408-420.

Résumé

For modern science, the study of geometries of local maximum mobility is of particular importance, including Euclidean and pseudo-Euclidean geometries, symplectic geometry, and geometries of constant curvature. There is no complete classification of such geometries at the exist. The author of this article developed a method, called the method of embedding, which makes it possible to carry out such a classification. The essence of this method consists in finding functions that define geometries of dimension $n+1$ using known functions that define geometries of dimension $n$. In this case, the desired function as an argument contains a known function of dimension geometry $n$ and two more variables. In addition, the requirement of local invariance of this function with respect to the transformation group with $(n+1)(n+2)/2 $ parameters is imposed. Then the condition of local invariance is written, from which the functional-differential equation is derived to the desired function. In this paper, the solutions of this equation are sought analytically, in the form of Taylor row. The problem formulated for pseudo-Euclidean geometry has three classes of solutions (geometries of local maximum mobility): pseudo-Euclidean geometry, special expansion of pseudo-Euclidean geometries, geometry on the pseudo sphere. In this paper we pose the embedding problem for special extensions of pseudo-Euclidean geometries. It is proved that the solutions of this problem are not the geometries of the local maximum mobility.

Détail

Keywords: functional equation, differential equation, metric function, geometry.

Bibliographie

[1] Mikhailichenko G.G., “Group and phenomenological symmetries in geometry”, Siberian Mathematical Journal, 25:5 (1984), 764–774 | DOI | MR | Zbl

[2] Mikhailichenko G.G., The mathematical basics and results of the theory of physical structures], Gorno-Altaysk State University, Gorno-Altaisk, 2016, 296 pp. (In Russ.)

[3] Mikhailichenko G.G., “Two-dimentional geometries”, Reports of the USSR Academy of Science, 24:2 (1981), 346–348 (In Russ.) | MR | Zbl

[4] Mikhailichenko G.G., “Two-dimensional geometries”, Reports of the USSR Academy of Sciences, 24:2 (1981), 346–348 (In Russ.) | MR | Zbl

[5] Kyrov V.A., “Functional equations in pseudo-Euclidean geometry”, Siberian journal of industrial mathematics, 13:4 (2010), 38–51 (In Russ.) | MR

[6] Kyrov V.A., Mikhailichenko G.G., “The analytic method of the symplectic geometry embedding”, Siberian Electronic Mathematical Reports, 14 (2017), 657–672 (In Russ.) | MR | Zbl

[7] Ovsyannikov L.V., Group Analysis of Differential Equations, Academic Press, New York, 1982, 432 pp. | MR | Zbl

[8] Fikhtengolts G.M., A course of differential and integral calculus, v. 2, Fizmatgiz Publ., Moscow, 1963, 524 pp. (In Russ.)

[9] Dyakonov V.P., Maple 10/11/12/13/14 in mathematical calculations, DMK Press Publ., Moscow, 2011, 802 pp. (In Russ.)