Algebraic aspects of web geometry. (English) Zbl 1042.53007

This common work was written on the occasion of 70 years of web geometry and has the character of a survey paper devoted to the relationship between (local) properties of equivalence classes (germs) of webs formed by suitable foliations of a manifold and properties of isotopy classes of corresponding coordinatizing differentiable local loops. The main results are given without proofs, many references are mentioned. Because of the length restrictions, only the most important interactions of web geometry and algebra are discussed, namely famous closure conditions and loop identities; differentiable local loops, their special classes and corresponding Akivis binary-ternary algebras; \((n+1)\)-webs and local \(n\)-quasigroups; special classes of four-webs and related algebraic conditions.


53A60 Differential geometry of webs
17D99 Other nonassociative rings and algebras
20N05 Loops, quasigroups
22A30 Other topological algebraic systems and their representations
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