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

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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 s-smooth manifold of dimension 2r, 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.

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Authors and Affiliations

  1. Psagot, Israel

    M. A. Akivis

  2. Tver State University, Tver, Russia

    A. M. Shelekhov

Authors
  1. M. A. Akivis
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  2. A. M. Shelekhov
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Correspondence to M. A. Akivis.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 16, No. 1, pp. 13–38, 2010.

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Akivis, M.A., Shelekhov, A.M. Cartan–Laptev method in the theory of multidimensional three-webs. J Math Sci 177, 522–540 (2011). https://doi.org/10.1007/s10958-011-0477-5

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