Alternators of a fourth order local analytic loop and three webs of multidimensional surfaces. (English. Russian original) Zbl 0701.53027

Sov. Math. 33, No. 4, 13-18 (1989); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1989, No. 4(323), 12-16 (1989).
Let W be a three-web of surfaces of codimension r given on a 2r- dimensional differentiable manifold. It is well-known [see M. A. Akivis, Tr. Geom. Semin. 2, 7-31 (1969; Zbl 0244.53014)] that a sequence of invariant tensor fields is associated with W: the torsion tensor \(a=a^ i_{jk}\), the curvature tensor \(b=b^ i_{jkl}\) and the covariant derivatives \(_{1}C=_{1}C^ i_{jklm}\), \(_{2}=_{2}C^ i_{jklm}\), \(i,j,k,l,m=1,...,r\), of the latter tensor etc. The torsion and curvature tensors define respectively the principal parts of the commutator and the associator in a coordinate loop L associated with the web W. This allows to describe special classes of webs W, which are defined by special structures of the tensors a and b, by means of identities in L.
In the paper under review it is proved that the tensors \(_{1}\) and \(_{2}\) have the similar meaning. Namely, up to a sign, they define two alternators which are determined by the fourth order chunk of the Taylor expansions of functions defining L.
Reviewer: V.V.Goldberg.


53A60 Differential geometry of webs


Zbl 0244.53014