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Generalizations of Frobenius’ theorem on manifolds and subcartesian spaces. (English) Zbl 1136.58004
A Hausdorff differential space $$S$$ in the sense of Sikorski is called a subcartesian space if it is covered by open sets which are diffeomorphic to subsets of $$\mathbb{R}^ n$$. A generalized distribution on a manifold $$M$$ is a subset $$D$$ of $$TM$$ such that $$D_p=D\cap T_pM$$ is a subspace of $$T_pM$$ for each $$p\in M$$. The dimension of $$D_p$$ is called the rank of $$D$$ at $$p$$. A generalized distribution is smooth if it is locally spanned by smooth vector fields. An integral manifold of a distribution $$D$$ on $$M$$ is an immersed submanifold $$N$$ of $$M$$ such that $$T\iota_{NM}(T_qN)= D_{\iota(q)}$$ for every $$q\in N$$, where $$\iota_{NM}: N \hookrightarrow M$$ is the inclusion map. A distribution $$D$$ on $$M$$ is said to be integrable if there exists an integral manifold of $$D$$ containing $$p$$ for every $$p\in M$$. If $$D$$ is integrable, then every integral manifold of $$D$$ can be extended to a maximal integral manifold of $$D$$. Let $$\mathcal D$$ be the family of all vector fields on $$M$$ with values in $$D$$. The distribution $$D$$ is said to be involutive if the family $$\mathcal D$$ is closed under the Lie bracket of vector fields. An integrable distribution is involutive.
The Theorem of Frobenius states that a constant rank distribution $$D$$ on a manifold $$M$$ is integrable if it is involutive, while the Theorem of Sussmann states that a distribution $$D$$ on a manifold $$M$$ is integrable if and only if it is preserved by local one-parameter groups of local diffeomorphisms of $$M$$ generated by vector fields on $$M$$ with values in $$D$$. I. Kolář, P. W. Michor, and J. Slovák proved that a distribution $$D$$ on a manifold $$M$$ is integrable if it is involutive and its rank is constant on integral curves of vector fields on $$M$$ with values in $$D$$. Frobenius’ and Sussmann’s theorems are special cases of this result. A distribution $$D$$ spanned by a family $${\mathcal F}$$ of vector fields on $$M$$ is said to be involutive on an orbit $$O$$ of $${\mathcal F}$$ if the Lie bracket $$[X, Y](p)\in D_p$$ for every $$X,Y\in{\mathcal F}$$ and every $$p\in O$$.
In this paper, the author shows that if $${\mathcal F}$$ is a family of vector fields on a manifold or a subcartesian space $$M$$ spanning a distribution $$D$$, then 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 the Theorem of Kolář, Michor, and Slovák, and hence it implies Frobenius’ and Sussmann’s Theorems.
##### MSC:
 58A30 Vector distributions (subbundles of the tangent bundles) 58A40 Differential spaces
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