Journal of Mathematical Sciences

, 177:623 | Cite as

Differential-geometric structures on generalized Reidemeister and Bol three-webs



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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  1. 1.
    M. A. Akivis, “On three-webs of multidimensional surfaces,” Itogi Nauki Tekh. Ser. Probl. Geom. Tr. Geom. Sem., 2, 7–31 (1969).MathSciNetGoogle Scholar
  2. 2.
    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).MathSciNetGoogle Scholar
  3. 3.
    M. A. Akivis and A. M. Shelekhov, Algebra and Geometry of Multidimensional Three-Webs, Kluwer Academic, Dordrecht (1992).MATHGoogle Scholar
  4. 4.
    I. A. Batalin, “Quasigroup construction and first-class constraints,” J. Math. Phys., 22, No. 9, 1837–1849 (1981).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    V. V. Goldberg, “On (n + 1)-webs of multidimensional surfaces,” Dokl. Akad. Nauk SSSR, 210, No. 4, 756–759 (1973).MathSciNetGoogle Scholar
  6. 6.
    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.Google Scholar
  7. 7.
    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).Google Scholar
  8. 8.
    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).MathSciNetGoogle Scholar
  9. 9.
    G. G. Mikhailichenko, “Solution of functional equations in theory of physical structures,” Dokl. Akad. Nauk SSSR, 206, No. 5, 1056–1058 (1972).MathSciNetGoogle Scholar
  10. 10.
    P. O. Mikheev, “On loops of transformations,” Deposited at VINITI (1985), No. 4531–85.Google Scholar
  11. 11.
    P. O. Mikheev, “Quasigroups of transformations,” Quasigroups and Nonassociative algebras in Physics (Tr. Inst. Fiziki, Tartu), 66, 54–66 (1990).Google Scholar
  12. 12.
    A. I. Nesterov, “Quasigroup ideas in physics,” Quasigroups and Nonassociative algebras in Physics (Tr. Inst. Fiziki, Tartu), 66, 107–120 (1990).MathSciNetGoogle Scholar
  13. 13.
    G. A. Tolstikhina, “On associative smooth monoids,” in: Webs and Quasigroups, Tver (2002), pp. 53–59.Google Scholar
  14. 14.
    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).Google Scholar
  15. 15.
    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).Google Scholar
  16. 16.
    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).Google Scholar
  17. 17.
    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.Google Scholar
  18. 18.
    G. A. Tolstikhina and A. M. Shelekhov, “Generalized associativity in smooth groupoids,” Dokl. Ross. Akad. Nauk, 383, No. 1, 32–33 (2002).MathSciNetGoogle Scholar
  19. 19.
    G. A. Tolstikhina and A. M. Shelekhov, “Three-webs defined by groups of transformations,” Dokl. Ross. Akad. Nauk, 385, No. 4, 1–3 (2002).MathSciNetGoogle Scholar
  20. 20.
    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.Google Scholar
  21. 21.
    G. A. Tolstikhina and A. M. Shelekhov, “On Bol quasigroups of transformations,” Dokl. Ross. Akad. Nauk, 401, No. 2, 166–168 (2005).MathSciNetGoogle Scholar
  22. 22.
    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).Google Scholar

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© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Tver State UniversityTverRussia

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