Geodesic
Closed $G$-structures defined by three-webs
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 153 (2011) no. 3, pp. 22-28 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We introduce the notion of $1$-digit identity of order $k$ and prove the theorem: if in coordinate loops of analytic three-web $W$ there hold $k-1$ independent identities of order $k$, $G$-structure defined by this web is a closed structure of class not higher than $2k$.
Keywords: multidimensional three-web, closed $G$-structure, coordinate loop of a three-web, $1$-digit identity of $k$-th order. 
@article{UZKU_2011_153_3_a1,
     author = {M. A. Akivis and A. M. Shelekhov},
     title = {Closed $G$-structures defined by three-webs},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {22--28},
     year = {2011},
     volume = {153},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2011_153_3_a1/}
}
TY  - JOUR
AU  - M. A. Akivis
AU  - A. M. Shelekhov
TI  - Closed $G$-structures defined by three-webs
JO  - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
PY  - 2011
SP  - 22
EP  - 28
VL  - 153
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UZKU_2011_153_3_a1/
LA  - ru
ID  - UZKU_2011_153_3_a1
ER  - 
%0 Journal Article
%A M. A. Akivis
%A A. M. Shelekhov
%T Closed $G$-structures defined by three-webs
%J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
%D 2011
%P 22-28
%V 153
%N 3
%U http://geodesic.mathdoc.fr/item/UZKU_2011_153_3_a1/
%G ru
%F UZKU_2011_153_3_a1
M. A. Akivis; A. M. Shelekhov. Closed $G$-structures defined by three-webs. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 153 (2011) no. 3, pp. 22-28. http://geodesic.mathdoc.fr/item/UZKU_2011_153_3_a1/

[1] Akivis M. A., “O zamknutykh $G$-strukturakh na differentsiruemom mnogoobrazii”, Itogi nauki i tekhniki. Problemy geometrii, 7, VINITI AN SSSR, M., 1975, 69–79 | MR | Zbl

[2] Santilli R. M., “Status of the mathematical and physical studies in the Lie-admissible formulations on July 1979 with particular references to the strong interactions”, Hadronic J., 2:6 (1979), 1460–2018 | MR

[3] Maltsev A. I., “Analiticheskie lupy”, Matem. sb., 36(78):3 (1955), 569–576 | MR | Zbl

[4] Holmes J. P., “Differentiable power associative groupoids”, Pacif. J. Math., 42:2 (1972), 391–394 | DOI | MR

[5] Holmes J. P., Sagle A. A., “Analytic $H$-spaces, Campbell-Hausdorff formula, and alternative algebras”, Pacific J. Math., 91:1 (1980), 105–134 | DOI | MR

[6] Ya. Lykhmus, P. Kuusk (red.), Kvazigruppy i neassotsiativnye algebry v fizike, Cb. st., Trudy In-ta fiziki, 66, Tartu, 1990, 235 pp. | MR

[7] Shelekhov A. M., “Klassifikatsiya mnogomernykh tpi-tkanei po usloviyam zamykaniya”, Itogi nauki i tekhniki. Problemy geometrii, 21, VINITI AN SSSR, M., 1989, 109–154 | MR | Zbl

[8] Akivis M. A., “O tri-tkanyakh mnogomernykh poverkhnostei”, Itogi nauki i tekhniki. Trudy geom. seminara, 2, VINITI AN SSSR, M., 1969, 7–31 | MR | Zbl

[9] Akivis M., Shelekhov A. M., Geometry and Algebra of Multidimensional Three-Webs, Kluwer Acad. Publ., Dordrecht–Boston–London, 1992, XVII+358 pp. | MR | Zbl

[10] Akivis M. A., Shelekhov A. M., Mnogomernye tri-tkani i ikh prilozheniya, Tver. gos. un-t, Tver, 2010, 307 pp.

[11] Shelekhov A. M., “The $g$-structure associated with a multidimensional hexagonal 3-web, is closed”, J. Geom., 35:1–2 (1989), 167–176 | DOI | MR | Zbl

[12] Shelekhov A. M., “O diffepentsialno-geometpicheskikh ob'ektakh vysshikh poryadkov mnogomernoi tpi-tkani”, Itogi nauki i tekhniki. Problemy geometrii, 19, VINITI AN SSSR, M., 1987, 101–154 | MR | Zbl

[13] Billig V. A., Shelekhov A. M., “O klassifikatsii tozhdestv s odnoi pepemennoi v gladkoi lokalnoi lupe”, Tkani i kvazigpuppy, Kalinin. gos. un-t, Kalinin, 1987, 24–32 | MR | Zbl

[14] Billig V. A., Shelekhov A. M., “Klassifikatsiya tozhdestv dliny 12 poryadka 4 s odnoi peremennoi v lokalnoi analiticheskoi lupe”, Tkani i kvazigpuppy, Kalinin. gos. un-t, Kalinin, 1990, 10–18 | MR

[15] Akivis M. A., “O kanonicheskikh razlozheniyakh uravnenii lokalnoi analiticheskoi kvazigruppy”, Dokl. AN SSSR, 188:5 (1969), 967–970 | MR | Zbl