Analytic embedding of some two-dimensional geometries of maximal mobility

V. A. Kyrov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 916-937.

Résumé

In this paper, we solve the problem of embedding two-dimensional geometries: simplicial, dual-gelmgoltz, Helmholtz proper and pseudohelmholtz, into three-dimensional geometries. This problem is solved by an analytical method. The functions defining these geometries are found. Basic operators of Lie algebras of groups of motions are calculated.

Détail

Keywords: geometry of maximum mobility, functional equation, differential equation, Lie algebra.
Mots-clés : Lie transformation group

Bibliographie

[1] G.G. Mikhailichenko, The mathematical basics and results of the theory of physical structures, 2012, arXiv: 1602.02795 [math-ph]

[2] G.G. Mikhailichenko, “Two-dimensional geometry”, Soviet Math. Doklady, 24:2 (1981), 346–348 | MR | Zbl

[3] H. Helmholz, “Uber die Thatsachen, die der Geometrie zu Grunde liegen”, Nachr. Konigl.Ges. Wiss. und Georg-Augusts-Univ. Gottingen, 1868, 19

[4] V.A. Kyrov, “Functional equations in pseudo-Euclidean geometry”, Sib. Zn. Ind. Mat, 13:4 (2010), 38–51 (in Russian) | MR | Zbl

[5] V.A. Kyrov, G.G. Mikhailichenko, “An analytic method for the embedding of the Euclidean and pseudo-Euclidean geometries”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 23, no. 2, 2017, 167–181 (in Russian) | MR

[6] V.A. Kyrov, “Ob odnom klasse funktsionalno-differentsialnykh uravnenii”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 26:1 (2012), 31–38 (in Russian) | DOI | MR

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

[8] V.A. Kyrov, “The analytical method for embedding multidimensional pseudo-Euclidean geometries”, Siberian Electronic Mathematical Reports, 15 (2018), 741–-758 (in Russian) | MR | Zbl

[9] G.G. Mikhailichenko, “On group and phenomenological symmetry in geometry”, Siberian Mathematical Journal, 25:5 (1984), 99–113 | MR | Zbl

[10] L.V. Ovsyannikov, A group analysis of differential equation, Nauka, M., 1978 | MR

[11] V.A. Kyrov, The geometry of Helmgolz, LAP, Saarbrucken, 2011

[12] G.M. Fichtengolz, A course of differential and integral calculus, v. 2, Phismhatgis, M., 1963 | MR

[13] L.E. Elsgoltz, Differential equations and calculus of variations, Nauka, M., 1969 | MR

[14] V.A. Kyrov, R.A. Bogdanova, “The groups of motions of some three-dimensional maximal mobility geometries”, Siberian Mathematical Journal, 59:2 (2018), 323–331 (in Russian) | DOI | MR | Zbl

[15] V.A. Kyrov, “Six-dimensional lie algebras of groups of motions of Tree-dimensional phenomenologically symmetric geometries”, Polymetric geometries, Novosibirsk State University, Novosibirsk, 2001, 116–143

[16] V. Dyakonov, Maple 10/11/12/13/14 in mathematical calculations, DMS, M., 2014

[17] V.H. Lev, “Three-dimensional geometries in the theory of physical structures”, Vychisl. Sistemy, 125 (1988), 90–103 (in Russian) | MR | Zbl

[18] V.A. Kyrov, “Embedding of phenomenologically symmetric geometries of two sets of the rank $(N,2)$ into phenomenologically symmetric geometries of two sets of the rank $(N+1,2)$”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 26:3 (2016), 312–323 (in Russian) | DOI | MR | Zbl

[19] V.A. Kyrov, “Embedding of phenomenologically symmetric geometries of two sets of the rank $(N,M)$ into phenomenologically symmetric geometries of two sets of the rank $(N+1,M)$”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 27:1 (2017), 42–53 (in Russian) | DOI | MR | Zbl