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 one specified by 1-forms alone, but the theory includes other types of example of differential system.

Given a collection of differential 1-forms on an n-dimensional manifold M, an integral submanifold is an embedding

of a submanifold N into M 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.

An integrability condition is a condition on the α_i to guarantee that there will be an integral submanifold.

Example of a non-integrable system

Not every such differential system has integral manifolds, however. For example, consider the following one-form on the standard simplex :

θ = xdy + ydz + zdx

Suppose that N is an integral submanifold for θ, so that i ^* θ = 0. In particular, i ^* dθ = di ^* θ = 0. So dθ is also in the kernel of i ^* , which means that we must have for some 1-form α on M. On the other hand, by the skewness of the wedge product, this implies that

But a direct calculation verifies that

which is a nonzero multiple of the standard volume on the simplex S, and so is never zero.

Necessary and sufficient conditions

The necessary and sufficient conditions for integrability of a system generated by 1-forms are supplied by the Frobenius theorem. One form states that if the ideal algebraically generated by the collection of α_i inside the ring Ω(M) is differentially closed , then the system admits an integral manifold.

Examples

In Riemannian geometry, we may consider the problem of finding an orthogonal coframe θⁱ (i.e., collection of 1-forms forming a basis of the cotangent space at every point with ) which are closed . By the Poincare 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 . 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

Φ = MΘ

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

On the other hand,

But ω = (dM)M ^{- 1} 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 which 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 0-387-97441-3
• Olver, P., "Equivalence, Invariants, and Symmetry," Cambridge, ISBN 0-521-47811-1