On four-dimensional three-webs with integrable transversal distributions

Abstract

For a four-dimensional (nonisoclinicly geodesic) three-web W (3, 2, 2), a transversal distribution $Δ$ is defined by the torsion tensor of the web. In general, this distribution is not integrable. The authors find necessary and sufficient conditions of its integrability and prove the existence theorem for webs W (3, 2, 2) with integrable distributions $Δ$. They prove that for a web W (3, 2, 2) with the integrable distribution $Δ$, the integral surfaces $V^2$ of $Δ$ are totally geodesic in an affine connection of a certain bundle of affine connections. They also consider a class of webs W (3, 2, 2) for which the integral surfaces $V^2$ of $Δ$ are geodesicly parallel with respect to the same affine connections and a class of webs for which two-dimensional webs W (3, 2, 1) cut by the foliations of W (3, 2, 2) on $V^2$ are hexagonal. They prove the existence theorems for webs of the latter class as well as for webs of the subclass which is the intersection of two classes mentioned above. The authors also establish relations between three-webs considered in the paper.