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Foliating Metric Spaces: A Generalization of Frobenius' Theorem

Commun. Math. Anal.
Volume 11, Number 1 (2011), 1 - 40

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 L2(R)is controllable by two elementary flows.