On One Class of Bol Three-Webs

Abstract

In this paper, we consider infinitesimal properties of multidimensional median Bol threewebs with a covariantly constant curvature tensor (webs ${B}_{m}^{\nabla }$) and lay the foundations for classifying such webs by the rank of the torsion tensor. For three-webs ${B}_{m}^{\nabla }$ of rank ρ, we construct an adapted frame by the Cartan method and find the corresponding system of structure (differential) equations. We prove that a three-web ${B}_{m}^{\nabla }$ of rank ρ carries a normal subweb, which is a group web, and the corresponding factor-web is a regular three-web. By integrating the structure equations, we find new families of examples of multidimensional three-webs of a special type and smooth Bol loops, which are a generalization of the semidirect product of two Abelian Lie groups.