Nonholonomic $(n+1)$-webs

On an $n$-manifold $M$ we consider nonholonomic $(n+1)$-web $NW$, which consists of $n+1$ distributions of codimension 1. We prove that the web $NW$ is equivalent to $G$-structure with structure group $\lambda E$, the group of scalar matrices. We obtain structure equations of the nonholonomic web $NW$ and find the integrability conditions of all its distributions. We show that a connection $\Gamma$ arises on the manifold $M$ carrying the web $NW$. Distributions of the web $NW$ are totally geodesic with respect to this connection. We consider the special case when the curvature of $\Gamma$ equals zero and in particular when the $(n+1)$-web $NW$ is formed by invariant distributions on the Lie group. We find the equations of the group when all distributions of $NW$ are integrable.