Smooth distributions are finitely generated

TL;DR

This paper proves that all smooth distributions on manifolds are finitely generated by a finite set of vector fields, but their sections may not form a finitely generated module over smooth functions, with results applicable to general vector bundles.

Contribution

It establishes the finite generation of smooth distributions and extends the results to arbitrary vector bundles, clarifying the structure of sections.

Findings

All smooth distributions are finitely generated.

The space of sections may not be finitely generated as a module.

Results hold for arbitrary vector bundles.

Abstract

A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle.