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 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 kth order in their coordinate loops and find the conditions for these webs to have the closed G-structure.

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Last modified: March 11, 2011