On differential-geometric structures on a manifold of nonholonomic (n + 1)-web

Abstract

We consider a nonholonomic (n + 1)-web NW on an n-dimensional manifold M, i.e., n + 1 distributions of codimension 1. We prove that there exists an invariant pencil of projective connections on M. To an ordered nonholonomic (n + 1)-web on M, there corresponds a unique curvilinear (n + 1)-web on M and vice versa. This correspondence is defined by the polarity with respect to a certain invariant multilinear n-form or the barycentric subdivision of some (n - 1)- dimensional simplex. In conclusion, we consider, as a special case, a nonholonomic (n + 1)-web ANW of hyperplanes in the affine space. A web ANW generates an invariant pencil of affine connections. We also study the case when these connections are projective.