Commun. Math. Anal.

Volume 11, Number 1 (2011), 1 - 40

Foliating Metric Spaces: A Generalization of Frobenius' Theorem

Foliating Metric Spaces: A Generalization of Frobenius' Theorem

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### 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(R)is controllable by two elementary flows.