ON ELASTIC 3-WEBS WITH CLASS 1 TORSION TENSOR

Dzhukashev Kamil' Ramilevich, Postgraduate student, Tver State University
(33 Zhelyabova street, Tver, Russia), dzhukashev@gmail.com

514.763.7

Background. Multidimensional 3-webs, produced on the smooth manifold by three layerings ofsimilar dimensionality, are the geometric interpretation of the function of two variables and have multiple applications, for example, in the theory of differential equations, in theoretical physics and in the theory of quasi-groups and loops. One of the least studied classes of 3-webs are the elastic 3-webs (E webs), isotopically corresponding to the invariant class of loops with the elasticity identity (xy)x = x(yx). The study is aimed at investigating elastic 3-webs, the algebra of which, derived from the algebra determined by the torsion tensor, is a one-dimensional one.
Materials and methods. To research E 3-webs the author uses the Elie Cartan method of external forms and moving frames, modified by G.F. Laptev. The article describes the usage of structural equations obtained by the said method.
Results. The author discovered a system of structural equations, that determines the class of webs, proved its closure in relation to external differentiation. Thus, it is proved that nontrivial elastic webs with class 1 torsion tensor exist. The researcher discovered correlations to tensors of the said web and proved the existence of an adaptive frame, in which the turvature tensor also has class 1, and the algebra, derived from the algebra determined by the torsion tensor, is in the same one-dimesional space.
Conclusion. the method of Cartan – Laptev allows effective researching of special classes of multidimensional 3-webs.

multidimensional 3-webs, Elie Cartan frame, method of Cartan - Laptev.

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