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.
Mathematical formulation 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 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 (not necessarily embedded) submanifold whose tangent space at every point p ∈ N {\displaystyle \textstyle p\in N} is annihilated by (the pullback of) each α i {\displaystyle \textstyle \alpha _{i}} . A maximal integral manifold is an immersed (not necessarily embedded) submanifold
i : N ⊂ M {\displaystyle i:N\subset M}
such that the kernel of the restriction map on forms
i ∗ : Ω p 1 ( M ) → Ω p 1 ( N ) {\displaystyle i^{*}:\Omega _{p}^{1}(M)\rightarrow \Omega _{p}^{1}(N)}
is spanned by the α i {\displaystyle \textstyle \alpha _{i}} at every point p {\displaystyle p} of N {\displaystyle N} . If in addition the α i {\displaystyle \textstyle \alpha _{i}} are linearly independent, then N {\displaystyle N} is ( n − k {\displaystyle n-k} )-dimensional. A Pfaffian system is said to be completely integrable if M {\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 α i {\displaystyle \alpha _{i}} to guarantee that there will be integral submanifolds of sufficiently high dimension.
Intuition
A Pfaffian system is specified by 1-forms. At each point x ∈ M {\displaystyle x\in M} , the set of 1-forms can be visualized as a set of hyperplanes, or contact elements, centered on the point. The hyperplanes intersect, producing a linear subspace of the local tangent space T x M {\displaystyle T_{x}M} . This field of linear subspaces locally look like infinitesimal pieces of a maximal integral manifold, but it might be impossible to put together these infinitesimal pieces into a maximal integral manifold. The pieces might twist against each other, breaking any attempt to piece them together. For example, if M {\displaystyle M} has 3 dimensions, then a single 1-form produces a field of planes, while two 1-forms that are linearly independent at every point produces a field of lines. Integrating a field of lines is always possible, but integrating a field of planes may be impossible, due to "twisting". Locally, such non-integrable field of planes look like the standard contact structure on R 3 {\displaystyle \mathbb {R} ^{3}} , defined by the 1-form d z − y d x {\displaystyle dz-ydx} .
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 I {\displaystyle {\mathcal {I}}} algebraically generated by the collection of αi inside the ring Ω(M) is differentially closed, in other words
d I ⊂ I , {\displaystyle d{\mathcal {I}}\subset {\mathcal {I}},}
then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.)
Examples
Integrable regular systems Given any regular foliation, we can simply take its differentials to obtain an integrable regular system. The rank of the system is the codimension of the foliation. The Hopf fibration is a foliation of the 3-sphere into circles, which is a regular foliation of codimension 2.
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