# zbMATH — the first resource for mathematics

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.

##### MSC:
 53C30 Differential geometry of homogeneous manifolds 20N05 Loops, quasigroups 22A22 Topological groupoids (including differentiable and Lie groupoids)
##### Keywords:
homogeneous space; quasigroup; loop