Foliating Metric Spaces: A Generalization of Frobenius' Theorem

Abstract

Using families of curves to generalize vector fields, the Lie bracket is defined on a metric space, $M$. For $M$ complete, versions of the local and global Frobenius theorems hold, and flows are shown to commute if and only if their bracket is zero. An example is given showing $L^{2}\left( \mathbb{R}\right) $ is controllable by two elementary flows.