$3$-dimensional Bol loops as sections in non-solvable Lie groups

Abstract

Our aim in this paper is to classify the $3$-dimensional connected differentiable global Bol loops, which have a non-solvable group as the group topologically generated by their left translations and to describe their relations to metric space geometries. The classification of global differentiable Bol loops significantly differs from the classification of local differentiable Bol loops. We treat the differentiable Bol loops as images of global differentiable sections $\sigma :G/H \to G$ such that for all $r,s \in \sigma (G/H)$ the element $rsr$ lies in $\sigma (G/H)$, where $H$ is the stabilizer of the identity $e$ of $L$ in $G$.