Integrability conditions for differential systems

In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example, including Lax pairs of integrable systems. A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold (which one sets equal to 0 to find solutions to the system).

Given a collection of differential 1-forms $\textstyle \alpha _{i},i=1,2,\dots ,k$ on an $\textstyle n$ -dimensional manifold ⁠ $M$ ⁠, an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point $\textstyle p\in N$ is annihilated by (the pullback of) each ⁠ $\textstyle \alpha _{i}$ ⁠.

A maximal integral manifold is an immersed (not necessarily embedded) submanifold

i:N\subset M

such that the kernel of the restriction map on forms

i^{*}:\Omega _{p}^{1}(M)\rightarrow \Omega _{p}^{1}(N)

is spanned by the $\textstyle \alpha _{i}$ at every point $p$ of ⁠ $N$ ⁠. If in addition the $\textstyle \alpha _{i}$ are linearly independent, then $N$ is (⁠ $n-k$ ⁠)-dimensional.

A Pfaffian system is said to be completely integrable if $M$ admits a foliation by maximal integral manifolds. (Note that the foliation need not be regular; i.e. the leaves of the foliation might not be embedded submanifolds.)

An integrability condition is a condition on the $\alpha _{i}$ to guarantee that there will be integral submanifolds of sufficiently high dimension.

Necessary and sufficient conditions

The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal ${\mathcal {I}}$ algebraically generated by the collection of α_i inside the ring Ω(M) is differentially closed, in other words

d{\mathcal {I}}\subset {\mathcal {I}},

then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.)

Example of a non-integrable system

Not every Pfaffian system is completely integrable in the Frobenius sense. For example, consider the following one-form on R³ ∖ (0,0,0):

\theta =z\,dx+x\,dy+y\,dz.

If dθ were in the ideal generated by θ we would have, by the skewness of the wedge product

\theta \wedge d\theta =0.

But a direct calculation gives

\theta \wedge d\theta =(x+y+z)\,dx\wedge dy\wedge dz,

which is a nonzero multiple of the standard volume form on R³. Therefore, there are no two-dimensional leaves, and the system is not completely integrable.

On the other hand, for the curve defined by

x=t,\quad y=c,\quad z=e^{-t/c},\qquad t>0

then θ defined as above is 0, and hence the curve is easily verified to be a solution (i.e. an integral curve) for the above Pfaffian system for any nonzero constant c.

Examples of applications

In pseudo-Riemannian geometry, we may consider the problem of finding an orthogonal coframe θⁱ, i.e., a collection of 1-forms that form a basis of the cotangent space at every point with $\langle \theta ^{i},\theta ^{j}\rangle =\delta ^{ij}$ that are closed (dθⁱ = 0, i = 1, 2, ..., n). By the Poincaré lemma, the θⁱ locally will have the form dxⁱ for some functions xⁱ on the manifold, and thus provide an isometry of an open subset of M with an open subset of Rⁿ. Such a manifold is called locally flat.

This problem reduces to a question on the coframe bundle of M. Suppose we had such a closed coframe

\Theta =(\theta ^{1},\dots ,\theta ^{n}).

If we had another coframe ⁠ $\Phi =(\phi ^{1},\dots ,\phi ^{n})$ ⁠, then the two coframes would be related by an orthogonal transformation

\Phi =M\Theta

If the connection 1-form is ω, then we have

d\Phi =\omega \wedge \Phi

On the other hand,

{\begin{aligned}d\Phi &=(dM)\wedge \Theta +M\wedge d\Theta \\&=(dM)\wedge \Theta \\&=(dM)M^{-1}\wedge \Phi .\end{aligned}}

But $\omega =(dM)M^{-1}$ is the Maurer–Cartan form for the orthogonal group. Therefore, it obeys the structural equation ⁠ $d\omega +\omega \wedge \omega =0$ ⁠, and this is just the curvature of M: $\Omega =d\omega +\omega \wedge \omega =0.$ After an application of the Frobenius theorem, one concludes that a manifold M is locally flat if and only if its curvature vanishes.

Generalizations

Many generalizations exist to integrability conditions on differential systems thar are not necessarily generated by one-forms. The most famous of these are the Cartan–Kähler theorem, which only works for real analytic differential systems, and the Cartan–Kuranishi prolongation theorem. See § Further reading for details. The Newlander–Nirenberg theorem gives integrability conditions for an almost-complex structure.