Generic distributions and Lie algebras of vector fields (Q685821)

The purpose of the present paper is to prove the following theorem: Let \({\mathcal D}(r,n)\) denote the space of germs at 0 of rank \(r\) distributions on \(\mathbb{R}^ n\). Put the Whitney topology on \({\mathcal D}(r,n)\). Let \(\Sigma \subset{\mathcal D}(r,n)\) denote the subset consisting of those distributions which admit some frame which generates a finite dimensional Lie algebra. Suppose that \(r(n-r)>n\). Then the complement of \(\Sigma\) is the countable intersection of open dense sets. In particular this complement is dense. From this theorem it follows that the generic rank \(r\) distribution on an \(n\)-dimensional manifold does not admit a frame of vector fields which generates a finite dimensional Lie algebra, provided \(r(n-r)>n\).