Geometry of Smooth Bol and Moufang Loops

Abstract

The aim of this Chapter is to prove that any local Ck smooth (k ≥ 3) Bol loop can be considered as a geodesic loop of a certain affinely connected manifold with zero curvature (with absolute parallelism). Further, the techniques of differential geometry is used to prove the main structure theorem in the theory of local analytic Bol loops. the affine connection in the question is a generalization of the Cartan connection with zero curvature on Lie groups.