On a class of three-webs of type \(W(4,4,2)\) with constant components of the fundamental tensor. (Q1589399)

\textit{M. A. Akivis} and \textit{V. V. Goldberg} [Tr. Geom. Semin. 4, 179--204 (1973; Zbl 0314.53011)] initiated the study of multidimensional three-webs \(W (p, q, r)\) formed by three foliations of dimensions \(p, q\) and \(r, p \geq q \geq r\), on a differentiable manifold \(M\) of dimension \(n = p + q\). In his previous papers [see in (*) The geometry of imbedded manifolds (Russian), Moskov. Gos. Ped. Inst., Moscow, 101--112 (1986; Zbl 0655.53014); in Webs and quasigroups (Russian), Kalinin. Gos. Univ., Kalinin, 82--87 (1987; Zbl 0617.53023)], the author considered the case \(p = q > r\), that is, he studied the webs \(W (p, p, r), p >r\). In particular he considered in [(*) loc. cit.] such webs \(W (p, p, r)\) for which the components of the basic tensor of \(W\) are constant on a subbundle \(\widetilde{M}'\) of a specialized frame bundle \(M'\) of \(M\). The bundle \(M'\) is a \(G\)-structure on \(M\), and the subbundle \(\widetilde{M}'\) is a \(G_0\)-structure, \(G_0 \subseteq G\). In the paper under review the author distinguishes a class of webs \(W (4, 4, 2)\) for which the group \(G_0\subseteq \text{GL} (2)\) is the affine group. On leaves of the third foliation of such a web, an affine structure is defined.