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Local differentiable quasigroups and webs. (English) Zbl 0737.53015
Quasigroups and loops: theory and applications, Sigma Ser. Pure Math. 8, 263-311 (1990).
[For the entire collection see Zbl 0704.00017.]
The paper presents the most important results in multidimensional web theory obtained since the 1930s up to now. Here \(d\)-webs \(W(d,n,r)\) formed by \(d\) foliations \(\lambda_ \alpha\) of codimension \(r\) in an open domain \(D\) of a differentiable manifold \(X^{nr}\) are considered, in particular, 3-webs \((d=3, n=2)\), \((n+1)\)-webs \(W(n+1,n,r,)\), and special classes of 4-webs \(W(4,2,r)\). The author discusses only classical subjects connected with “the quasigroup point of view” for webs, namely (1) the closure conditions of different configurations on webs; (2) structure equations and tensor characterizations of web classes; (3) local coordinate loops, their canonical expansions and their tangent algebras. Besides the classical web types defined by the closure conditions, isoclinic and transversally geodesic webs (especially Grassmannian and algebraic webs) introduced by M. A. Akivis [Tr. Geom. Semin. 2, 7-31 (1969; Zbl 0244.53014)] are considered.

MSC:
53A60 Differential geometry of webs
20N05 Loops, quasigroups
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry