Cartan--Laptev method in the theory of multidimensional three-webs

M. A. Akivis ; A. M. Shelekhov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 1, pp. 13-38.

Résumé

We show how the Cartan–Laptev method which generalizes Elie Cartan's method of external forms and moving frames is supplied to the study of closed $G$-structures defined by multidimensional three-webs formed on a $C^s$-smooth manifold of dimension $2r$, $r\ge1$, $s\ge3$, by a triple of foliations of codimension $r$. We say that a tensor $T$ belonging to a differential-geometric object of order $s$ of three-web $W$ is closed if it can be expressed in terms of components of objects of lower order $s$. We find all closed tensors of a three-web and the geometric sense of one of relations connecting three-web tensors. We also point out some sufficient conditions for the web to have a closed $G$-structure. It follows from our results that the $G$-structure associated with a hexagonal three-web $W$ is a closed $G$-structure of class 4. It is proved that basic tensors of a three-web $W$ belonging to a differential-geometric object of order $s$ of the web can be expressed in terms of $s$-jet of the canonical expansion of its coordinate loop, and conversely. This implies that the canonical expansion of every coordinate loop of a three-web $W$ with closed $G$-structure of class $s$ is completely defined by an $s$-jet of this expansion. We also consider webs with one-digit identities of $k$th order in their coordinate loops and find the conditions for these webs to have the closed $G$-structure.

Bibliographie

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

[2] Akivis M. A., “O tri-tkanyakh mnogomernykh poverkhnostei”, Itogi nauki i tekhn. Ser. Probl. geom. Tr. geom. sem., 2, 1969, 7–31 | MR | Zbl

[3] Akivis M. A., “O zamknutykh $G$-strukturakh na differentsiruemom mnogoobrazii”, Itogi nauki i tekhn. Ser. Probl. geom. Tr. geom. sem., 7, 1975, 69–79 | MR | Zbl

[4] Akivis M. A., Shelekhov A. M., “O vychislenii tenzorov krivizny i krucheniya mnogomernoi tri-tkani i assotsiatora svyazannoi s nei lokalnoi kvazigruppy”, Sib. mat. zhurn., 12:5 (1971), 953–960 | MR | Zbl

[5] Akivis M. A., Shelekhov A. M., “O kanonicheskikh koordinatakh v lokalnoi analiticheskoi lupe”, Tkani i kvazigruppy, Kalininskii gos. un-t, Kalinin, 1986, 120–124 | MR | Zbl

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

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

[8] Evtushik L. E., Lumiste Yu. G., Ostianu N. M., Shirokov A. P., “Differentsialno-geometricheskie struktury na mnogoobraziyakh”, Itogi nauki i tekhn. Ser. Probl. geom. Tr. geom. sem., 9, 1979, 5–246 | MR | Zbl

[9] Kvazigruppy i neassotsiativnye algebry v fizike, Tartu, 1990 | MR

[10] Kulakov Yu. I., Elementy teorii fizicheskikh struktur, S dobavleniem G. G. Mikhailichenko, Novosibirsk, 1968

[11] Kulakov Yu. I., Vladimirov Yu. S., Karnaukhov A. V., Vvedenie v teoriyu fizicheskikh struktur i binarnuyu geometrofiziku, Arkhimed, M., 1992

[12] Lumiste Yu. G., “Svyaznosti v odnorodnykh rassloeniyakh”, Mat. sb., 69(111):3 (1966), 434–469 | MR | Zbl

[13] Lumiste Yu. G., “Matrichnoe predstavlenie polugolonomnoi differentsialnoi gruppy i strukturnye uravneniya rassloeniya $p$-koreperov”, Itogi nauki i tekhn. Ser. Probl. geom. Tr. geom. sem., 5, 1974, 239–257 | MR | Zbl

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

[15] Mikhailichenko G. G., “Reshenie funktsionalnykh uravnenii v teorii fizicheskikh struktur”, DAN SSSR, 206:5 (1972), 1056–1058 | Zbl

[16] Shelekhov A. M., O zamknutykh $g$-strukturakh, opredelyaemykh mnogomernymi tri-tkanyami, Dep. v VINITI 25.12 1985, No 8815-V, Kalinin, 49 pp.

[17] Shelekhov A. M., “O vychislenii kovariantnykh proizvodnykh tenzora krivizny mnogomernoi tri-tkani”, Tkani i kvazigruppy, Kalininskii gos. un-t, Kalinin, 1986, 96–103 | MR | Zbl

[18] Shelekhov A. M., “O differentsialno-geometricheskikh ob'ektakh vysshikh poryadkov mnogomernoi tri-tkani”, Itogi nauki i tekhn. Ser. Probl. geom. Tr. geom. sem., 19, 1987, 101–154 | MR | Zbl

[19] Shelekhov A. M., “Klassifikatsiya mnogomernykh tri-tkanei po usloviyam zamykaniya”, Itogi nauki i tekhn. Ser. Probl. geom. Tr. geom. sem., 21, 1989, 109–154 | MR | Zbl

[20] Shelekhov A. M., “Vychislenie vtorykh kovariantnykh proizvodnykh tenzora krivizny mnogomernoi tri-tkani”, Tkani i kvazigruppy, Kalininskii gos. un-t, Kalinin, 1990, 49–55 | MR | Zbl

[21] Akivis M. A., Shelekhov A. M., Geometry and Algebra of Multidimensional Three-Webs, Kluwer Academic, Dordrecht, 1992 | MR | Zbl

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

[23] 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

[24] Santilli R. M., Lie-Admissible Approach to the Hadronic Structure, v. I, Hadronic Press, Palm Harbor, 1978 | MR

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