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Fundamentalnaya i Prikladnaya Matematika, 2010, Volume 16, Issue 1, Pages 157–169 (Mi fpm1298)  

This article is cited in 1 scientific paper (total in 1 paper)

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

G. A. Tolstikhina

Tver State University
Full-text PDF (630 kB) Citations (1)
References:
Abstract: In this paper, we present the main results of the study of multidimensional three-webs $W(p,q,r)$ 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 $p=\lambda l$, $q=\lambda m$, and $r=\lambda(l+m-1)$. For such webs, we define the notion of a generalized Reidemeister configuration and proved that a three-web $W(\lambda l,\lambda m,\lambda (l+m-1))$, on which all sufficiently small generalized Reidemeister configurations are closed, are generated by a $\lambda$-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 acting 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 studies.
English version:
Journal of Mathematical Sciences (New York), 2011, Volume 177, Issue 4, Pages 623–632
DOI: https://doi.org/10.1007/s10958-011-0488-2
Bibliographic databases:
Document Type: Article
UDC: 514.763.7
Language: Russian
Citation: G. A. Tolstikhina, “Differential-geometric structures on generalized Reidemeister and Bol three-webs”, Fundam. Prikl. Mat., 16:1 (2010), 157–169; J. Math. Sci., 177:4 (2011), 623–632
Citation in format AMSBIB
\Bibitem{Tol10}
\by G.~A.~Tolstikhina
\paper Differential-geometric structures on generalized Reidemeister and Bol three-webs
\jour Fundam. Prikl. Mat.
\yr 2010
\vol 16
\issue 1
\pages 157--169
\mathnet{http://mi.mathnet.ru/fpm1298}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2786499}
\transl
\jour J. Math. Sci.
\yr 2011
\vol 177
\issue 4
\pages 623--632
\crossref{https://doi.org/10.1007/s10958-011-0488-2}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80052261932}
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  • https://www.mathnet.ru/eng/fpm/v16/i1/p157
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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