Abstract
In this paper, we present the main results of the study of multidimensional three-websW(p, q, r) obtained by the method of external forms and moving Cartan frame. The method was developed by the Russian mathematicians S. P. Finikov, G. F. Laptev, and A. M. Vasiliev, while fundamentals of differential-geometric (p, q, r)-webs theory were described by M. A. Akivis and V. V. Goldberg. Investigation of (p, q, r)-webs, including algebraic and geometric theory aspects, has been continued in our papers, in particular, we found the structure equations of a three-web W(p, q, r), where p = λl, q = λm, and r = λ(l + m − 1). For such webs, we define the notion of a generalized Reidemeister configuration and proved that a three-web W(λl, λm, λ(l + m − 1)), on which all sufficiently small generalized Reidemeister configurations are closed, is generated by a λ-dimensional Lie group G. The structure equations of the web are connected with the Maurer–Cartan equations of the group G. We define generalized Reidemeister and Bol configurations for three-webs W(p, q, q). It is proved that a web W(p, q, q) on which generalized Reidemeister or Bol configurations are closed is generated, respectively, by the action of a local smooth q-parametric Lie group or a Bol quasigroup on a smooth p-dimensional manifold. For such webs, the structure equations are found and their differential-geometric properties are studied.
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References
M. A. Akivis, “On three-webs of multidimensional surfaces,” Itogi Nauki Tekh. Ser. Probl. Geom. Tr. Geom. Sem., 2, 7–31 (1969).
M. A. Akivis and V. V. Goldberg, “On multidimensional three-webs formed by surfaces of different dimensions,” Dokl. Akad. Nauk SSSR, 203, No. 2, 263–266 (1972).
M. A. Akivis and A. M. Shelekhov, Algebra and Geometry of Multidimensional Three-Webs, Kluwer Academic, Dordrecht (1992).
I. A. Batalin, “Quasigroup construction and first-class constraints,” J. Math. Phys., 22, No. 9, 1837–1849 (1981).
V. V. Goldberg, “On (n + 1)-webs of multidimensional surfaces,” Dokl. Akad. Nauk SSSR, 210, No. 4, 756–759 (1973).
V. V. Goldberg, “Transversal-geodesic, hexagonal and group three-webs formed on surfaces of different dimensions,” in: Sb. Stat. po Differents. Geom. [in Russian], Kalinin (1974), pp. 52–64.
V. V. Goldberg, “On reducible group and (2n + 2)-hedron (n + 1)-webs of multidimensional surfaces,” Sib. Mat. Zh., 17, No. 1, 44–57 (1976).
J. Lykhmus, E. Paal, and L. Sorgsepp, “Nonassociativity in mathematics and physics,” Quasigroups and Nonassociative algebras in Physics (Tr. Inst. Fiziki, Tartu), 66, 8–22 (1990).
G. G. Mikhailichenko, “Solution of functional equations in theory of physical structures,” Dokl. Akad. Nauk SSSR, 206, No. 5, 1056–1058 (1972).
P. O. Mikheev, “On loops of transformations,” Deposited at VINITI (1985), No. 4531–85.
P. O. Mikheev, “Quasigroups of transformations,” Quasigroups and Nonassociative algebras in Physics (Tr. Inst. Fiziki, Tartu), 66, 54–66 (1990).
A. I. Nesterov, “Quasigroup ideas in physics,” Quasigroups and Nonassociative algebras in Physics (Tr. Inst. Fiziki, Tartu), 66, 107–120 (1990).
G. A. Tolstikhina, “On associative smooth monoids,” in: Webs and Quasigroups, Tver (2002), pp. 53–59.
G. A. Tolstikhina, “Algebra and geometry of three-webs formed by foliations of different dimensions,” Itogi Nauki Tekh. Ser. Sovrem. Mat. Ee Pril., 32, 29–116 (2005).
G. A. Tolstikhina, “To geometry of smooth mappings R q × R p → R λ generalizing groups,” Vestn. Tverskogo Gos. Univ. Ser. Prikl. Mat., Vyp. 5, No. 11 (39), 19–38 (2007).
G. A. Tolstikhina, “On local symmetric structure connected with generalized left Bol three-web B l (p, q, q),” Geometry, Topology and Their Applications, Sb. Rab. Inst. Mat. NAN Ukrainy, 6, No. 2, 247–255 (2009).
G. A. Tolstikhina and A. M. Shelekhov, “On three-webs W(p, q, p + q − 1) generalized Reidemeister configurations are closed on,” Deposited at VINITI (13.08.2001), No. 1869-V2001.
G. A. Tolstikhina and A. M. Shelekhov, “Generalized associativity in smooth groupoids,” Dokl. Ross. Akad. Nauk, 383, No. 1, 32–33 (2002).
G. A. Tolstikhina and A. M. Shelekhov, “Three-webs defined by groups of transformations,” Dokl. Ross. Akad. Nauk, 385, No. 4, 1–3 (2002).
G. A. Tolstikhina and A. M. Shelekhov, “Embedding of three-web defined by group of transformations in a group three-web,” Deposited at VINITI (2003), No. 880-V2003.
G. A. Tolstikhina and A. M. Shelekhov, “On Bol quasigroups of transformations,” Dokl. Ross. Akad. Nauk, 401, No. 2, 166–168 (2005).
G. A. Tolstikhina and A. M. Shelekhov, “On Bol three-web formed by foliations of different dimensions,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5 (516), 56–62 (2005).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 16, No. 1, pp. 157–169, 2010.
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Tolstikhina, G.A. Differential-geometric structures on generalized Reidemeister and Bol three-webs. J Math Sci 177, 623–632 (2011). https://doi.org/10.1007/s10958-011-0488-2
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DOI: https://doi.org/10.1007/s10958-011-0488-2