FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 1, PAGES 13-38

**Cartan--Laptev method in the theory of multidimensional three-webs**

M. A. Akivis

A. M. Shelekhov

Abstract

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We show how the Cartan--Laptev method which generalizes Elie Cartan's
method of external forms and moving frames is supplied to the study of
closed $G$-structures defined by
multidimensional three-webs formed on a $Cs$-smooth
manifold of dimension $2r$, $r$³
1, $s$³ 3, by a triple
of foliations of codimension $r$.
We say that a tensor $T$ belonging to
a differential-geometric object of order $s$ of
three-web $W$ is closed if it can be
expressed in terms of components of objects of lower
order $s$.
We find all closed tensors of a three-web and the geometric sense
of one of relations connecting three-web tensors.
We also point out some sufficient conditions for the web to have a
closed $G$-structure.
It follows from our results that the $G$-structure associated with
a hexagonal three-web $W$ is a closed $G$-structure of
class $4$.
It is proved that basic tensors of a three-web $W$ belonging to a
differential-geometric object of order $s$ of the web can be
expressed in terms of $s$-jet of the canonical
expansion of its coordinate loop, and conversely.
This implies that the canonical expansion of every coordinate loop of
a three-web $W$ with closed $G$-structure of
class $s$ is
completely defined by an $s$-jet of this expansion.
We also consider webs with one-digit identities of $k$th order in their
coordinate loops and find the conditions for these webs to have the
closed $G$-structure.

Location: http://mech.math.msu.su/~fpm/eng/k10/k101/k10102h.htm

Last modified: March 11, 2011