Frobenius theorem

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In mathematics, Frobenius' theorem states that a subbundle of the tangent bundle of a manifold is integrable if and only if it arises from a regular foliation.

Frobenius' theorem is one of the supporting pillars of differential topology and the calculus on manifolds; the only if part, an existence theorem for foliations, turns analysis into geometry.

The theorem is named after Ferdinand Georg Frobenius, but according to H. B. Lawson, it is actually due to Alfred Clebsch and F. Deahna.

Contents

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Introduction

The Frobenius theorem relates integrability to foliation; to state the theorem, both concepts must be clearly defined.

One begins by noting 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. Indeed, vector fields are often defined to be the derivatives of a collection of smooth curves.

This idea of integrability can be extended to collections of vector fields as well. One says that a subbundle <math>E\subset TM</math> of the tangent bundle TM is integrable, if, for any two vector fields X and Y taking values in E, then the Lie bracket <math>[X,Y]</math> takes values in E as well. This notion of integrability need only be defined locally; that is, the existence of the vector fields X and Y and their integrability need only be defined on subsets of M.

A subbundle <math>E\subset TM</math> may also be defined to arise from a foliation of a manifold. Let <math>N\subset M</math> be a submanifold that is a leaf of a foliation. Consider the tangent bundle TN. If TN is exactly E restricted to N, then one says that E arises from a regular foliation of M. Again, this definition is purely local: the foliation is defined only on charts.

Given the above definitions, Frobenius' theorem states that a subbundle E is integrable if and only if it arises from a regular foliation of M.

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Formal definition

A more formal and more abstract definition follows.

Let U be an open set in a manifold M, Ω1(U) be the space of smooth, differentiable 1-forms on U, and F be a submodule of Ω1(U) of rank r, the rank being constant in value over U. The Frobenius theorem states that F is integrable if and only if for every <math>p\in U</math> the stalk Fp is generated by r exact differential forms.

Geometrically, the theorem states that an integrable module of 1-forms of rank r is the same thing as a codimension-r foliation. The correspondence to the definition in terms of vector fields given in the introduction follows from the close relationship between differential forms and Lie derivatives. Frobenius' theorem 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 complex manifold (manifolds over <math>\mathbb{C}</math>) as well as real manifolds. It can be generalized to differential forms of higher degree and, in some instances, to the singular case.

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See also

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References

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