Basis of trancendense in differential field of invariants of pseugo-Galilean group

K. K. Muminov ; V. I. Chilin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2019), pp. 19-31.

Résumé

Let $G$ be a subgroup of the group of invertible linear operators, acting in a finite-dimensional real linear space $X$. One of the problems of differential geometry is the study of necessary and sufficient conditions for $G$-equivalence of paths in $X$. In solving this problem we use methods of the theory of differential invariants describing transcendence basis of differential fields of $G$-invariant differential rational functions. Using explicit descriptions of these bases allows to establish criteria for $G$-equivalence of paths in $X$. Such an approach has been used in solving the problem of equivalence of paths with respect to the actions of special linear, orthogonal, pseudo-orthogonal, and symplectic groups. We give an explicit description of one of the finite bases of transcendence in the differential field of invariant differential rational functions with respect to the action of the pseudo-Galilean group $\mathrm{\Gamma }O$. Using this basis, necessary and sufficient conditions are established for the $\mathrm{\Gamma }O$-equivalence of paths in $X$.

Détail

Mots-clés : pseudo-Galilean space, group of movements, differential rational function, transcendence basis, differential invariant, regular path.

Bibliographie

[1] Kartan E., Teoriya konechnykh nepreryvnykh grupp i differentsialnaya geometriya, izlozhennaya metodom podvizhnogo repera, PLATON, Volgograd, 1998

[2] Kartan E., Izbrannye trudy, MTsNMO, M., 1998

[3] Aminov Yu. A., Differentsialnaya geometriya i topologiya, Nauka, M., 1987

[4] Blyashke V., Differentsialnaya geometriya i geometricheskie osnovy teorii otnositelnosti Einshteina, ONTI, M–L., 1935

[5] Shirokov P. A., Shirokov A. P., Affinnaya differentsialnaya geometriya, Fiz.-mat.lit., M., 1959 | MR

[6] Pommaret J. F., Differential Galois theory, Mathematics and its Appl., 15, Gordon Breach Science Publishers, New York, 1983 | MR | Zbl

[7] Aripov R. G., Khadzhiev Dzh., “Polnaya sistema globalnykh differentsialnykh i integralnykh invariantov krivoi v evklidovoi geometrii”, Izv. vuzov. Matem., 2007, no. 7, 1–14 | MR | Zbl

[8] Muminov K. K., “Ekvivalentnost putei otnositelno deistviya simplekticheskoi gruppy”, Izv. vuzov. Matem., 2002, no. 7, 27–38

[9] Muminov K. K., Chilin V. I., Ekvivalentnost krivykh v konechnomernykh vektornykh prostranstvakh, LAP LAMBEPT Academic Publishing, Germaniya, 2015

[10] Chilin V. I., Muminov K. K., “Polnaya sistema differentsialnykh invariantov krivoi v psevdoevklidovom prostranstve”, Dinam. sistemy, 2013, no. 1–2, 135–149 | Zbl

[11] Khadjiev Dj., Peksen Ö., “The complete system of global integral and differential invariants for equi-affine curves”, Diff. Geom. Appl., 20:2 (2004), 167–175 | DOI | MR | Zbl

[12] Rozenfeld B. A., Neevklidovy prostranstva, Nauka, M., 1969

[13] Kaplanskii I., Vvedenie v differentsialnuyu algebru, In. lit., M., 1959

[14] Kolchin E. R., Differential algebra and algebraic groups, Academic Press, New York–London, 1973 | MR | Zbl

[15] Khadzhiev Dzh., Prilozhenie teorii invariantov k differentsialnoi geometrii krivykh, FAN, Tashkent, 1998

[16] Muminov K. K., Chilin V. I., “Bazisy transtsendentnosti v differentsialnykh polyakh invariantnykh ratsionalnykh funktsii”, Uzbeksk. matem. zhurn., 2016, no. 1, 85–96