Integrability Conditions For Differential Systems

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 one 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.