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The statement remains true for analytic 1-forms and in the holomorphic case, with C instead of R. It can be generalized to differential forms of higher degree and, in some instances, to the singular case.
There is also a statement in terms of vector fields, which makes the sufficient condition for the existence of a submanifold of U of codimension r, tangent to vector fields
- X1, X2, ..., Xr,
that the Lie bracket
of any two of the given fields should lie in the space spanned by them. Since the Lie bracket can equally be taken on the submanifold, this condition is certainly necessary. The relationship between the two aspects is because the Lie bracket and exterior derivative are connected.
This last restatement of the theorem can be taken to show the integrability of vector fields on the manifold. In this variant, one can state that an arbitrary smooth vector field X on a manifold M can be integrated to define a one-paramter family of curves. The integrability follows because the equation defining the curve is a first-order ordinary differential equation, and thus its integrability is gaurenteed by the Picard-Lindelöf theorem.
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