On a class of middle Bol three-web

Background. The middle Bol three-webs (three-webs $B_m^\bigtriangledown$) occupy a special place in the theory of webs. Their algebraic analogue is the multidimensional smooth Bol loops - following the analytic Moufang loops in the vicinity of their properties to Lie groups. Therefore, the classification of middle Bol three-webs is of great importance. The aim of this paper is to consider the infinitesimal properties of multidimensional middle Bol three-webs with a covariantly constant curvature tensor and show that the class of three-webs $B_m^\bigtriangledown$ with torsion tensor of rank 1 coincides with the class of elastic three-webs $E_1^r$. Materials and methods. The methods of tensor analysis, external differential calculus, the theory of connections, theory of Lie groups are applied in the article. The main method of investigation is the E. Cartan method of external forms and the moving frame, adapted by M. A. Akivis, V. V. Goldberg and A. M. Shelekhov for the study of the theory of multidimensional webs. The main material used is the structural theory of multidimensional three-webs, developed by M. A. Akivis, as well as the results of scientific research on the theory of multidimensional Bol three-webs. Results. It is proved that the class of three-webs $B_m^\bigtriangledown$ with torsion tensor of rank 1 coincides with the class of elastic three-webs $E_1^r$. Conclusions. The result obtained shows the necessity of investigating the Bol three-webs $B_m^\bigtriangledown$ with the torsion tensor of rank $\rho>1$.