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Cartan–Laptev method in the theory of multidimensional three-webs
M. A. Akivisa,
A. M. Shelekhovb a Israel
b Tver State University
Abstract:
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
$C^s$-smooth manifold of dimension
$2r$,
$r\ge1$,
$s\ge3$, 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.
UDC:
514.763.7
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