Webs and curvature (Q2782005)

For three-webs \(W (3, 2, r)\) defined by three foliations of codimension \(r\) on a differentiable manifold of dimension \(2r\), the author defines the canonical Chern connection and gives a complete global description of left invariant webs on Lie groups. He presents a proof of Blaschke's theorem that for a three-web \(W (3, 2, 1)\) in the plane all hexagonal figures are closed if and only if the web curvature vanishes. For three-webs \(W (3, 2, r)\), the author finds the relations between local closure conditions (Reidemeister, Moufang, Bol, hexagonal) and a) weak-associative identities in binary local differentiable loops associated with \(W (3, 2, r)\) and b) corresponding infinitesimal identities of the commutator and associator operations of their tangent algebras. In the last section, the author investigates special classes of webs \(W (3, 2, r)\). In particular, he proves that a web \(W (3, 2, r)\) is transversally geodesic if and only if it possesses a maximal family of 2-dimensional subwebs and that a web \(W (3, 2, r)\) is torsionless (or paratactical in Akivis' terminology) if and only if a certain distribution of \(r\)-planes is integrable.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00024].