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

View as HTML
View as gif image

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.

Location: http://mech.math.msu.su/~fpm/eng/k10/k101/k10113h.htm

Last modified: March 11, 2011