Generalizations of Frobenius’ theorem on manifolds and subcartesian spaces.

*(English)*Zbl 1136.58004A 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.

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.

Reviewer: Andrew Bucki (Edmond)