Affine homogeneous structures on analytic loops.

*(English)*Zbl 0647.53012Generalizing the notion of a homogeneous structure on a Riemannian manifold [see F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds (1983; Zbl 0509.53043)], the author introduces the general concept of an affine homogeneous structure and considers such structures on an analytic loop with the canonical affine connection [see the author, Hiroshima Math. J. 5, 141-179 (1975; Zbl 0304.53037)]. The main problem which is under investigation is to find homogeneous loops [see the author, loc. cit.] on a given analytic loop (G,\(\mu)\) by changing its multiplication \(\mu\) to a homogeneous one, provided that the unit e of \(\mu\) is unchanged and every 1-parameter subgroup of \(\mu\) is changed to a 1-parameter subgroup of the homogeneous loop obtained.

The loop (G,\(\mu)\) is said to be geodesic at e if it coincides with the geodesic local loop [see M. A. Akivis, Sib. Math. J. 19, 171-178 (1978); translation from Sib. Mat. Zh. 19, 243-253 (1978; Zbl 0388.53007)] of the canonical affine connection centered at e. If (G,\(\mu)\) is geodesic at e, then any geodesic curve c(t) passing through the unit \(e=c(0)\) is a 1-parameter subgroup of the loop. Let (G,\(\mu)\) be a connected and simply connected geodesic loop. If the canonical affine connection of (G,\(\mu)\) admits an affine homogeneous structure, then, by changing every 1-parameter subgroup of \(\mu\) to a 1-parameter subgroup of \(\mu\) ’, the loop (G,\(\mu)\) can be changed to a homogeneous loop (G,\(\mu\) ’). This result allows a criterion for the existence of a change of a homogeneous loop to a homogeneous one to be found in terms of the tangent Lie triple algebras.

The loop (G,\(\mu)\) is said to be geodesic at e if it coincides with the geodesic local loop [see M. A. Akivis, Sib. Math. J. 19, 171-178 (1978); translation from Sib. Mat. Zh. 19, 243-253 (1978; Zbl 0388.53007)] of the canonical affine connection centered at e. If (G,\(\mu)\) is geodesic at e, then any geodesic curve c(t) passing through the unit \(e=c(0)\) is a 1-parameter subgroup of the loop. Let (G,\(\mu)\) be a connected and simply connected geodesic loop. If the canonical affine connection of (G,\(\mu)\) admits an affine homogeneous structure, then, by changing every 1-parameter subgroup of \(\mu\) to a 1-parameter subgroup of \(\mu\) ’, the loop (G,\(\mu)\) can be changed to a homogeneous loop (G,\(\mu\) ’). This result allows a criterion for the existence of a change of a homogeneous loop to a homogeneous one to be found in terms of the tangent Lie triple algebras.

Reviewer: V.V.Goldberg

##### MSC:

53A60 | Differential geometry of webs |

53C30 | Differential geometry of homogeneous manifolds |

22A30 | Other topological algebraic systems and their representations |