FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 1, PAGES 157-169

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

G. A. Tolstikhina

Abstract

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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 $p\; =$l l, $q\; =$l m, and $r\; =$l (l+m - 1). For such webs, we define the notion of a generalized Reidemeister configuration and proved that a three-web $W($l l, l m, l (l+m - 1)), on which all sufficiently small generalized Reidemeister configurations are closed, are generated by a $$l-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.

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