1 Answers
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.
Given a collection of differential 1-forms α i , i = 1 , 2 , … , k {\displaystyle \textstyle \alpha _{i},i=1,2,\dots ,k} on an n {\displaystyle \textstyle n} -dimensional manifold M {\displaystyle M} , an integral manifold is an immersed submanifold whose tangent space at every point p ∈ N {\displaystyle \textstyle p\in N} is annihilated by each α i {\displaystyle \textstyle \alpha _{i}}.
A maximal integral manifold is an immersed submanifold
such that the kernel of the restriction map on forms