# 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. A **Pfaffian system** is specified by 1-forms alone, but the theory includes other types of example of **differential system**.

Given a collection of differential 1-forms α_{i}, *i*=1,2, ..., *k* on an *n*-dimensional manifold *M*, an **integral manifold** is a submanifold whose tangent space at every point *p* ∈ *M* is annihilated by each α_{i}.

A **maximal integral manifold** is a submanifold

such that the kernel of the restriction map on forms

is spanned by the α_{i} at every point *p* of *N*. If in addition the α_{i} are linearly independent, then *N* is (*n* − *k*)-dimensional. Note that *i*: *N* ⊂ *M* need not be an embedded submanifold.

A Pfaffian system is said to be **completely integrable** if *N* 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 α_{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 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 *d*θ 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, the curve defined by

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 M with an open subset of

^{i}**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.

## 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