Frobenius theorem

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In mathematics, the Frobenius theorem states, in its smooth version of degree 1, the following:

Let U be an open set in Rn and F a submodule of Ω1(U) of constant rank r in U. Then F is integrable if and only if for every p ∈ U the stalk Fp is generated by r exact differential forms.

Geometrically, it states that an integrable module of 1-forms of rank r is the same thing as a codimension-r foliation. It is one of the basic tools for the study of vector fields and foliations.

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

[Xi,Xj]

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-parameter family of curves. The integrability follows because the equation defining the curve is a first-order ordinary differential equation, and thus its integrability is guaranteed by the Picard-Lindelöf theorem.

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