Frobenius theorem (differential topology)

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.
Contact geometry studies 1-forms that maximally violate the assumptions of Frobenius' theorem. An example is shown on the right.

Introduction

One-form version

Suppose we are to find the trajectory of a particle in a subset of 3D space, but we do not know its trajectory formula. Instead, we know only that its trajectory satisfies, where are smooth functions of. Thus, our only certainty is that if at some moment in time the particle is at location, then its velocity at that moment is restricted within the plane with equation
In other words, we can draw a "local plane" at each point in 3D space, and we know that the particle's trajectory must be tangent to the local plane at all times.
If we have two equationsthen we can draw two local planes at each point, and their intersection is generically a line, allowing us to uniquely solve for the curve starting at any point. In other words, with two 1-forms, we can foliate the domain into curves.
If we have only one equation, then we might be able to foliate into surfaces, in which case, we can be sure that a curve starting at a certain surface must be restricted to wander within that surface. If not, then a curve starting at any point might end up at any other point in. One can imagine starting with a cloud of little planes, and quilting them together to form a full surface. The main danger is that, if we quilt the little planes two at a time, we might go on a cycle and return to where we began, but shifted by a small amount. If this happens, then we would not get a 2-dimensional surface, but a 3-dimensional blob. An example is shown in the diagram on the right.
If the one-form is integrable, then loops exactly close upon themselves, and each surface would be 2-dimensional. Frobenius' theorem states that this happens precisely when over all of the domain, where. The notation is defined in the article on one-forms.
During his development of axiomatic thermodynamics, Carathéodory proved that if is an integrable one-form on an open subset of, then for some scalar functions on the subset. This is usually called Carathéodory's theorem in axiomatic thermodynamics. One can prove this intuitively by first constructing the little planes according to, quilting them together into a foliation, then assigning each surface in the foliation with a scalar label. Now for each point, define to be the scalar label of the surface containing point.
Now, is a one-form that has exactly the same planes as. However, it has "even thickness" everywhere, while might have "uneven thickness". This can be fixed by a scalar scaling by, giving. This is illustrated on the right.

Multiple one-forms

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
be a collection of functions, with, and such that the matrix has rank r when evaluated at any point of. Consider the following system of partial differential equations for a function :
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
for, and all functions u, and for some coefficients c^k_ij 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
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 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 by:
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. Suppose that are solutions of the problem 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 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 are not unique, the equation nonetheless determines a unique family of level sets. Just as in the case of the example, general solutions u of are in a one-to-one correspondence with functions on the family of level sets.
The level sets corresponding to the maximal independent solution sets of 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 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 on a manifold defines a family of curves, its integral curves . These are the solutions of , which is a system of first-order ordinary differential equations, whose solvability is guaranteed by the Picard-Lindelöf theorem. If the vector field is nowhere zero then it defines a one-dimensional subbundle of the tangent bundle of, and the integral curves form a regular foliation of. 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 of the tangent bundle is integrable, if, for any two vector fields and taking values in, the Lie bracket takes values in as well. This notion of integrability need only be defined locally; that is, the existence of the vector fields and and their integrability need only be defined on subsets of.
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= : 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 =_α∈A.
Trivially, any foliation of defines an integrable subbundle, since if and is the leaf of the foliation passing through then is integrable. Frobenius' theorem states that the converse is also true:
Given the above definitions, Frobenius' theorem states that a subbundle is integrable if and only if the subbundle arises from a regular foliation of.

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 consists of all forms such that
for all. The set I forms a subring and, in fact, an ideal in. Furthermore, using the definition of the exterior derivative, it can be shown that I is closed under exterior differentiation 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.