Smooth distributions are globally finitely spanned

Abstract

A smooth distribution on a smooth manifold M is, by definition, a map that assigns to each point x of M a linear subspace Δ(x) of the tangent space T x M, in such a way that, locally, there exist smooth sections f1,., fd of Δ such that the linear span of f 1(x),., f d (x) is Δ(x) for all x. We prove that a much weaker definition of smooth distribution, in which it is only required that for each x M and each v Δ(x) there exist a smooth section f of Δ defined near x such that f(x)=v, suffices to imply that there exists a finite family {f 1,., f d } of smooth global sections of Δ such that Δ(x) is spanned, for every x M, by the values f 1(x),., f d (x). The result is actually proved for general singular subbundles E of an arbitrary smooth vector bundle V, and we give a bound on the number d of global spanning sections, by showing that one can always take d=rankE · (1 + dimM), where rankE is the maximum dimension of the fibers E(x). © 2008 Springer-Verlag.