Frobenius theorem (differential topology)

Solutions to underdetermined systems of equations are seldom unique. For example, the system of differential equations

${\displaystyle {\begin{cases}{\frac {\partial f}{\partial x}}+{\frac {\partial f}{\partial y}}=0\\{\frac {\partial f}{\partial y}}+{\frac {\partial f}{\partial z}}=0\end{cases}}}$

clearly permits multiple solutions. Nevertheless, these solutions still have enough structure that they may be completely described. The first observation is that, even if f₁ and f₂ are two different solutions, the level surfaces of f₁ and f₂ must overlap. In fact, the level surfaces for this system are all planes in R³ of the form x − y + z = C, for C 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:

${\displaystyle f(x,y,z)=C(t){\text{ whenever }}x-y+z=t.}$

Conversely, if a function C(t) 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 u₁, ..., u_n−r are solutions of the problem (1) satisfying the independence condition on the gradients. Consider the level sets[2] of (u₁, ..., u_n−r) as functions with values in R^n−r. If v₁, ..., v_n−r 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.[3]

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.

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 ${\displaystyle X}$ on a manifold ${\displaystyle M}$ defines a family of curves, its integral curves ${\displaystyle u:I\to M}$ (for intervals ${\displaystyle I}$). These are the solutions of ${\displaystyle {\dot {u}}(t)=X_{u(t)}}$, which is a system of first-order ordinary differential equations, whose solvability is guaranteed by the Picard–Lindelöf theorem. If the vector field ${\displaystyle X}$ is nowhere zero then it defines a one-dimensional subbundle of the tangent bundle of ${\displaystyle M}$, and the integral curves form a regular foliation of ${\displaystyle M}$. 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 ${\displaystyle E\subset TM}$ of the tangent bundle ${\displaystyle TM}$ is integrable (or involutive), if, for any two vector fields ${\displaystyle X}$ and ${\displaystyle Y}$ taking values in ${\displaystyle E}$, the Lie bracket ${\displaystyle [X,Y]}$ takes values in ${\displaystyle E}$ as well. This notion of integrability need only be defined locally; that is, the existence of the vector fields ${\displaystyle X}$ and ${\displaystyle Y}$ and their integrability need only be defined on subsets of ${\displaystyle M}$.

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 {L_α}_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=(x¹, ⋅⋅⋅, xⁿ) : U→Rⁿ 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 ${\displaystyle {\mathcal {F}}}$={L_α}_α∈A.[4]

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

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

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 p in U the stalk F_p 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.

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 Ω(M) of all forms on M. These two forms are related by duality. If D is a smooth tangent distribution on M, then the annihilator of D, I(D) consists of all forms ${\displaystyle \alpha \in \Omega ^{k}(M)}$ (for any ${\displaystyle k\in \{1,\dots ,\operatorname {dim} M\}}$) such that

${\displaystyle \alpha (v_{1},\dots ,v_{k})=0}$

for all ${\displaystyle v_{1},\dots ,v_{k}\in D}$. The set I(D) forms a subring and, in fact, an ideal in Ω(M). 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 I(D) is closed under exterior differentiation if and only if D is integrable.

One infinite-dimensional generalization is as follows.[5] Let X and Y be Banach spaces, and A ⊂ X, B ⊂ Y a pair of open sets. Let

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

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

${\displaystyle (1)\quad y'=F(x,y)}$

if

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

The equation (1) is completely integrable if for each ${\displaystyle (x_{0},y_{0})\in A\times B}$, there is a neighborhood U of x₀ such that (1) has a unique solution u(x) defined on U such that u(x₀)=y₀.

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

${\displaystyle D_{1}F(x,y)\cdot (s_{1},s_{2})+D_{2}F(x,y)\cdot (F(x,y)\cdot s_{1},s_{2})=D_{1}F(x,y)\cdot (s_{2},s_{1})+D_{2}F(x,y)\cdot (F(x,y)\cdot s_{2},s_{1})}$

for all s₁, s₂ ∈ X. Here D₁ (resp. D₂) denotes the partial derivative with respect to the first (resp. second) variable; the dot product denotes the action of the linear operator F(x, y) ∈ L(X, Y), as well as the actions of the operators D₁F(x, y) ∈ L(X, L(X, Y)) and D₂F(x, y) ∈ L(Y, L(X, Y)).

The infinite-dimensional version of the Frobenius theorem also holds on Banach manifolds.[6] The statement is essentially the same as the finite-dimensional version.

Let M be a Banach manifold of class at least C². Let E be a subbundle of the tangent bundle of M. The bundle E is involutive if, for each point p ∈ M and pair of sections X and Y of E defined in a neighborhood of p, the Lie bracket of X and Y evaluated at p, lies in E_p:

${\displaystyle [X,Y]_{p}\in E_{p}}$

On the other hand, E is integrable if, for each p ∈ M, there is an immersed submanifold φ : N → M whose image contains p, such that the differential of φ is an isomorphism of TN with φ⁻¹E.

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

The statement of the theorem remains true for holomorphic 1-forms on complex manifolds — manifolds over C with biholomorphic transition functions.[7]

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

${\displaystyle d\omega ^{j}=\sum _{i=1}^{r}\psi _{i}^{j}\wedge \omega ^{i}}$

for some system of holomorphic 1-forms ψ^j
_i, 1 ≤ i, j ≤ r, then there exist holomorphic functions f_i^j and gⁱ such that, on a possibly smaller domain,

${\displaystyle \omega ^{j}=\sum _{i=1}^{r}f_{i}^{j}dg^{i}.}$

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 Cⁿ is not restrictive.

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