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

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

${\displaystyle i:N\subset M}$

such that the kernel of the restriction map on forms

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

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

A Pfaffian system is said to be completely integrable if ${\displaystyle M}$ 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 ${\displaystyle \alpha _{i}}$ to guarantee that there will be integral submanifolds of sufficiently high dimension.

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 ${\displaystyle {\mathcal {I}}}$ algebraically generated by the collection of α_i inside the ring Ω(M) is differentially closed, in other words

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

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

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

${\displaystyle \theta =z\,dx+x\,dy+y\,dz.}$

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

${\displaystyle \theta \wedge d\theta =0.}$

But a direct calculation gives

${\displaystyle \theta \wedge d\theta =(x+y+z)\,dx\wedge dy\wedge dz}$

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

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

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.

In Riemannian geometry, we may consider the problem of finding an orthogonal coframe θⁱ, i.e., a collection of 1-forms forming a basis of the cotangent space at every point with ${\displaystyle \langle \theta ^{i},\theta ^{j}\rangle =\delta ^{ij}}$ which 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

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

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

${\displaystyle \Phi =M\Theta }$

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

${\displaystyle d\Phi =\omega \wedge \Phi }$

On the other hand,

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

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

Many generalizations exist to integrability conditions on differential systems which 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.

• Bryant, Chern, Gardner, Goldschmidt, Griffiths, Exterior Differential Systems, Mathematical Sciences Research Institute Publications, Springer-Verlag, ISBN 0-387-97411-3
• Olver, P., Equivalence, Invariants, and Symmetry, Cambridge, ISBN 0-521-47811-1
• Ivey, T., Landsberg, J.M., Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, American Mathematical Society, ISBN 0-8218-3375-8
• Dunajski, M., Solitons, Instantons and Twistors, Oxford University Press, ISBN 978-0-19-857063-9