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- Title
Differential-geometric structures on generalized Reidemeister and Bol three-webs.
- Authors
Tolstikhina, G.
- 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 = 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.
- Subjects
DIFFERENTIAL geometry; GENERALIZED spaces; REIDEMEISTER torsion; WEBS (Differential geometry); MATHEMATICAL forms; LIE groups; MANIFOLDS (Mathematics)
- Publication
Journal of Mathematical Sciences, 2011, Vol 177, Issue 4, p623
- ISSN
1072-3374
- Publication type
Article
- DOI
10.1007/s10958-011-0488-2