Smooth Distributions Are Globally Finitely Spanned

Abstract

Summary. 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).