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 on an -dimensional manifold , an **integral manifold** is an immersed (not necessarily embedded) submanifold whose tangent space at every point is annihilated by (the pullback of) each .

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

such that the kernel of the restriction map on forms

is spanned by the at every point of . If in addition the are linearly independent, then is ()-dimensional.

A Pfaffian system is said to be **completely integrable** if 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 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 algebraically generated by the collection of α_{i} inside the ring Ω(*M*) is differentially closed, in other words

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 **R**^{3} − (0,0,0):

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

But a direct calculation gives

which is a nonzero multiple of the standard volume form on **R**^{3}. Therefore, there are no two-dimensional leaves, and the system is not completely integrable.

On the other hand, for the curve defined by

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 Riemannian geometry, we may consider the problem of finding an orthogonal coframe *θ*^{i}, i.e., a collection of 1-forms forming a basis of the cotangent space at every point with which are closed (dθ^{i} = 0, *i* = 1, 2, ..., *n*). By the Poincaré lemma, the θ^{i} locally will have the form d*x ^{i}* for some functions

*x*on the manifold, and thus provide an isometry of an open subset of

^{i}*M*with an open subset of

**R**

^{n}. 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

If we had another coframe , then the two coframes would be related by an orthogonal transformation

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

On the other hand,

But is the Maurer–Cartan form for the orthogonal group. Therefore, it obeys the structural equation and this is just the curvature of M: 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 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.

## Further reading

- 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