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
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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 -structures defined by
multidimensional three-webs formed on a -smooth
manifold of dimension , , , by a triple
of foliations of codimension .
We say that a tensor belonging to
a differential-geometric object of order of
three-web is closed if it can be
expressed in terms of components of objects of lower
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
It follows from our results that the -structure associated with
a hexagonal three-web is a closed -structure of
It is proved that basic tensors of a three-web belonging to a
differential-geometric object of order of the web can be
expressed in terms of -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 with closed -structure of
completely defined by an -jet of this expansion.
We also consider webs with one-digit identities of th order in their
coordinate loops and find the conditions for these webs to have the
Last modified: March 11, 2011