Left Bol three-webs with the IC-property

Abstract

In this paper, we describe some properties of the surface V. For a Bol web Bwith the IC-property, we find expansions into a series (up to the third degree) of the equations of the local coordinate loop and the equations of the core. With the use of these expansions, it is proved that if a left Bol web Bhas the IC-property, then each its coordinate loopP , P ∈ V , has zero commutator. We find the equations of the submanifold V in terms of an adapted cobasis and the expansion into a series of the function y = ϕ(x) which determines the surface V in terms of canonical coordinates. We prove (Theorem 2) that the surface V is an isoclinic-geodesic and diagonal surface of the web W in question and that the existence of such surfaces characterizes webs Bwith theIC-property. It turns out that the conditions of existence of the submanifold V are equivalent to the existence of a submanifold on which (and, respectively, on the base of the first foliation of the web) the symmetric connection Γ coincides with the Chern canonical affine connection of B� . The results obtained are illustrated by examples.