Frobenius Integration Theorem

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of ''r'' vector fields mesh into coordinate grids on ''r''-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds.

Introduction

In its most elementary form, the theorem addresses the problem of finding a maximal set of independent solutions of a regular system of first-order linear homogeneous partial differential equations. Let

:$\left \$

be a collection of functions, with , and such that the matrix has rank ''r''. Consider the following system of partial differential equations for a function :

:$(1) \quad \begin L_1u\ \stackrel\ \sum_i f_1^i(x)\frac = 0\\ L_2u\ \stackrel\ \sum_i f_2^i(x)\frac = 0\\ \qquad \cdots \\ L_ru\ \stackrel\ \sum_i f_r^i(x)\frac = 0 \end$

One seeks conditions on the existence of a collection of solutions such that the gradients are linearly independent.

The Frobenius theorem asserts that this problem admits a solution locally if, and only if, the operators satisfy a certain integrability condition known as ''involutivity''. Specifically, they must satisfy relations of the form

:$L_iL_ju(x)-L_jL_iu(x)=\sum_k c_^k(x)L_ku(x)$

for , and all functions ''u'', and for some coefficients ''c''^''k''_''ij''(''x'') that are allowed to depend on ''x''. In other words, the commutators must lie in the linear span of the at every point. The involutivity condition is a generalization of the commutativity of partial derivatives. In fact, the strategy of proof of the Frobenius theorem is to form linear combinations among the operators so that the resulting operators do commute, and then to show that there is a coordinate system for which these are precisely the partial derivatives with respect to .

From analysis to geometry

Even though the system is overdetermined there are typically infinitely many solutions. For example, the system of differential equations

:$\begin \frac + \frac =0\\ \frac+ \frac=0 \end$

clearly permits multiple solutions. Nevertheless, these solutions still have enough structure that they may be completely described. The first observation is that, even if ''f''₁ and ''f''₂ are two different solutions, the level surfaces of ''f''₁ and ''f''₂ must overlap. In fact, the level surfaces for this system are all planes in of the form , for a constant. The second observation is that, once the level surfaces are known, all solutions can then be given in terms of an arbitrary function. Since the value of a solution ''f'' on a level surface is constant by definition, define a function ''C''(''t'') by:

:$f(x,y,z)=C(t) \text x - y + z = t.$

Conversely, if a function is given, then each function ''f'' given by this expression is a solution of the original equation. Thus, because of the existence of a family of level surfaces, solutions of the original equation are in a one-to-one correspondence with arbitrary functions of one variable.

Frobenius' theorem allows one to establish a similar such correspondence for the more general case of solutions of (1). Suppose that are solutions of the problem (1) satisfying the independence condition on the gradients. Consider the level sets of as functions with values in . If is another such collection of solutions, one can show (using some linear algebra and the mean value theorem) that this has the same family of level sets but with a possibly different choice of constants for each set. Thus, even though the independent solutions of (1) are not unique, the equation (1) nonetheless determines a unique family of level sets. Just as in the case of the example, general solutions ''u'' of (1) are in a one-to-one correspondence with (continuously differentiable) functions on the family of level sets.

The level sets corresponding to the maximal independent solution sets of (1) are called the ''integral manifolds'' because functions on the collection of all integral manifolds correspond in some sense to constants of integration. Once one of these constants of integration is known, then the corresponding solution is also known.

Frobenius' theorem in modern language

The Frobenius theorem can be restated more economically in modern language. Frobenius' original version of the theorem was stated in terms of Pfaffian systems, which today can be translated into the language of differential forms. An alternative formulation, which is somewhat more intuitive, uses vector fields.

Formulation using vector fields

In the vector field formulation, the theorem states that a subbundle of the tangent bundle of a manifold is integrable (or involutive) if and only if it arises from a regular foliation. In this context, 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 $M$ defines a family of curves, its integral curves $u:I\to M$ (for intervals $I$). These are the solutions of $\dot u(t) = X_$, which is a system of first-order ordinary differential equations, whose solvability is guaranteed by the Picard–Lindelöf theorem. If the vector field $X$ is nowhere zero then it defines a one-dimensional subbundle of the tangent bundle of $M$, and the integral curves form a regular foliation of $M$. Thus, one-dimensional subbundles are always integrable.

If the subbundle has dimension greater than one, a condition needs to be imposed.
One says that a subbundle $E\subset TM$ of the tangent bundle $TM$ is integrable (or involutive), if, for any two vector fields $X$ and $Y$ taking values in $E$, the Lie bracket ,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 $M$.

Several definitions of foliation exist. Here we use the following:

Definition. A ''p''-dimensional, class ''C^r'' foliation of an ''n''-dimensional manifold ''M'' is a decomposition of ''M'' into a union of disjoint connected submanifolds _α∈''A'', called the ''leaves'' of the foliation, with the following property: Every point in ''M'' has a neighborhood ''U'' and a system of local, class ''C^r'' coordinates ''x''=(''x''¹, ⋅⋅⋅, ''xⁿ'') : ''U''→R^''n'' such that for each leaf ''L''_α, the components of ''U'' ∩ ''L''_α are described by the equations ''x''^''p''+1=constant, ⋅⋅⋅, ''xⁿ''=constant. A foliation is denoted by $\mathcal$=_α∈''A''.

Trivially, any foliation of $M$ defines an integrable subbundle, since if $p\in M$ and $N\subset M$ is the leaf of the foliation passing through $p$ then $E_p = T_pN$ is integrable. Frobenius' theorem states that the converse is also true:

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

Differential forms formulation

Let ''U'' be an open set in a manifold , be the space of smooth, differentiable 1-forms on ''U'', and ''F'' be a submodule of 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 in the stalk ''F_p'' is generated by ''r'' exact differential forms.

Geometrically, the theorem states that an integrable module of -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.

There are thus two forms of the theorem: one which operates with distributions, that is smooth subbundles ''D'' of the tangent bundle ''TM''; and the other which operates with subbundles of the graded ring of all forms on ''M''. These two forms are related by duality. If ''D'' is a smooth tangent distribution on , then the annihilator of ''D'', ''I''(''D'') consists of all forms $\alpha\in\Omega^k (M)$ (for any $k\in \$) such that

:$\alpha(v_1,\dots,v_k) = 0$

for all $v_1,\dots,v_k\in D$. The set ''I''(''D'') forms a subring and, in fact, an ideal in . Furthermore, using the definition of the exterior derivative, it can be shown that ''I''(''D'') is closed under exterior differentiation (it is a differential ideal) if and only if ''D'' is involutive. Consequently, the Frobenius theorem takes on the equivalent form that is closed under exterior differentiation if and only if ''D'' is integrable.

Generalizations

The theorem may be generalized in a variety of ways.

Infinite dimensions

One infinite-dimensional generalization is as follows. Let and be Banach spaces, and a pair of open sets. Let

:$F:A\times B \to L(X,Y)$

be a continuously differentiable function of the Cartesian product (which inherits a differentiable structure from its inclusion into ''X'' × ''Y'') into the space of continuous linear transformations of into ''Y''. A differentiable mapping ''u'' : ''A'' → ''B'' is a solution of the differential equation

:$(1) \quad y' = F(x,y)$

if

:$\forall x \in A: \quad u'(x) = F(x, u(x)).$

The equation (1) is completely integrable if for each $(x_0, y_0)\in A\times B$, there is a neighborhood ''U'' of ''x''₀ such that (1) has a unique solution defined on ''U'' such that ''u''(''x''₀)=''y''₀.

The conditions of the Frobenius theorem depend on whether the underlying field is or . If it is R, then assume ''F'' is continuously differentiable. If it is , then assume ''F'' is twice continuously differentiable. Then (1) is completely integrable at each point of if and only if

:$D_1F(x,y)\cdot(s_1,s_2) + D_2F(x,y)\cdot(F(x,y)\cdot s_1,s_2) = D_1F(x,y) \cdot (s_2,s_1) + D_2F(x,y)\cdot(F(x,y)\cdot s_2,s_1)$

for all . Here (resp. ) denotes the partial derivative with respect to the first (resp. second) variable; the dot product denotes the action of the linear operator , as well as the actions of the operators and .

Banach manifolds

The infinite-dimensional version of the Frobenius theorem also holds on Banach manifolds. The statement is essentially the same as the finite-dimensional version.

Let be a Banach manifold of class at least ''C''². Let be a subbundle of the tangent bundle of . The bundle is involutive if, for each point and pair of sections and ''Y'' of defined in a neighborhood of ''p'', the Lie bracket of and ''Y'' evaluated at ''p'', lies in :

: ,Yp \in E_p

On the other hand, is integrable if, for each , there is an immersed submanifold whose image contains ''p'', such that the differential of is an isomorphism of ''TN'' with .

The Frobenius theorem states that a subbundle is integrable if and only if it is involutive.

Holomorphic forms

The statement of the theorem remains true for holomorphic 1-forms on complex manifolds — manifolds over with biholomorphic transition functions.

Specifically, if $\omega^1,\dots,\omega^r$ are ''r'' linearly independent holomorphic 1-forms on an open set in such that

:$d\omega^j = \sum_^r \psi_i^j \wedge \omega^i$

for some system of holomorphic 1-forms , then there exist holomorphic functions ''f''_i^j and such that, on a possibly smaller domain,

:$\omega^j = \sum_^r f_i^jdg^i.$

This result holds locally in the same sense as the other versions of the Frobenius theorem. In particular, the fact that it has been stated for domains in is not restrictive.

Higher degree forms

The statement does not generalize to higher degree forms, although there is a number of partial results such as Darboux's theorem and the Cartan-Kähler theorem.

History

Despite being named for Ferdinand Georg Frobenius, the theorem was first proven by Alfred Clebsch and Feodor Deahna. Deahna was the first to establish the sufficient conditions for the theorem, and Clebsch developed the necessary conditions. Frobenius is responsible for applying the theorem to Pfaffian systems, thus paving the way for its usage in differential topology.

Applications

* In classical mechanics, the integrability of a system's constraint equations determines whether the system is holonomic or nonholonomic.

See also

* Integrability conditions for differential systems
* Domain-straightening theorem
* Newlander-Nirenberg Theorem

Notes

References

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Theorems in differential geometry
Theorems in differential topology
Differential systems
Foliations