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, 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.