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Homogeneous spaces and quasigroups. (English. Russian original) Zbl 0874.53037
Russ. Math. 40, No. 7, 74-81 (1996); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1996, No. 7(410), 77-84 (1996).
The paper is a review dedicated to loops and homogeneous spaces. Let \(G/H\) be a homogeneous space, \(Q\) be a submanifold of the Lie group \(G\) through the unit of \(G\) transversal to the cosets \(gH\). Then we can define a binary operation on \(Q\) by projection on \(Q\) along the leaves \(gH\). Conversely, one can reconstruct a homogeneous space \(G/H\) from the given loop \(Q\). The connection between \(Q\) and \(G/H\) gives a possibility to describe the homogeneous space properties in terms of quasigroup and loop theory.

53C30 Differential geometry of homogeneous manifolds
20N05 Loops, quasigroups
22A22 Topological groupoids (including differentiable and Lie groupoids)