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Cartan-Laptev method in the theory of multidimensional three-webs.
- Source :
-
Journal of Mathematical Sciences . Sep2011, Vol. 177 Issue 4, p522-540. 19p. - Publication Year :
- 2011
-
Abstract
- We show how the Cartan-Laptev method that generalizes Elie Cartan's method of external forms and moving frames is applied to the study of closed G-structures defined by multidimensional three-webs formed on a C-smooth manifold of dimension 2 r, r ≥ 1, s ≥ 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 a 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 an 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 10723374
- Volume :
- 177
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Journal of Mathematical Sciences
- Publication Type :
- Academic Journal
- Accession number :
- 65101804
- Full Text :
- https://doi.org/10.1007/s10958-011-0477-5