Let $\mathcal{F}$ be a family of vector fields on a manifold or a subcartesian space spanning a distribution $D$. We prove that an orbit $O$ of $\mathcal{F}$ is an integral manifold of $D$ if $D$ is involutive on $O$ and it has constant rank on $O$. This result implies Frobenius' theorem, and its various generalizations, on manifolds as well as on subcartesian spaces.

Keywords:

differential spaces, generalized distributions, orbits, Frobenius' theorem, Sussmann's theorem differential spaces, generalized distributions, orbits, Frobenius' theorem, Sussmann's theorem

MSC Classifications:

58A30, 58A40 show english descriptions Vector distributions (subbundles of the tangent bundles)
Differential spaces 58A30 - Vector distributions (subbundles of the tangent bundles)
58A40 - Differential spaces

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