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
style\alphai,i=1,2,...,k
stylen
M
stylep\inN
style\alphai
A maximal integral manifold is an immersed (not necessarily embedded) submanifold
i:N\subsetM
such that the kernel of the restriction map on forms
1(M) → | |
i | |
p |
1(N) | |
\Omega | |
p |
is spanned by the
style\alphai
p
N
style\alphai
N
n-k
A Pfaffian system is said to be completely integrable if
M
An integrability condition is a condition on the
\alphai
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
lI
d{lI}\subset{lI},
then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.)
Not every Pfaffian system is completely integrable in the Frobenius sense. For example, consider the following one-form :
\theta=zdx+xdy+ydz.
If dθ were in the ideal generated by θ we would have, by the skewness of the wedge product
\theta\wedged\theta=0.
But a direct calculation gives
\theta\wedged\theta=(x+y+z)dx\wedgedy\wedgedz
which is a nonzero multiple of the standard volume form on R3. Therefore, there are no two-dimensional leaves, and the system is not completely integrable.
On the other hand, for the curve defined by
x=t, y=c, z=e-t/c, t>0
then θ defined as above is 0, and hence the curve is easily verified to be a solution (i.e. an integral curve) for the above Pfaffian system for any nonzero constant c.
In Riemannian geometry, we may consider the problem of finding an orthogonal coframe θi, i.e., a collection of 1-forms forming a basis of the cotangent space at every point with
\langle\thetai,\thetaj\rangle=\deltaij
This problem reduces to a question on the coframe bundle of M. Suppose we had such a closed coframe
\Theta=(\theta1,...,\thetan).
If we had another coframe
\Phi=(\phi1,...,\phin)
\Phi=M\Theta
If the connection 1-form is ω, then we have
d\Phi=\omega\wedge\Phi
On the other hand,
\begin{align} d\Phi&=(dM)\wedge\Theta+M\wedged\Theta\\ &=(dM)\wedge\Theta\\ &=(dM)M-1\wedge\Phi. \end{align}
But
\omega=(dM)M-1
d\omega+\omega\wedge\omega=0,
\Omega=d\omega+\omega\wedge\omega=0.
Many generalizations exist to integrability conditions on differential systems which are not necessarily generated by one-forms. The most famous of these are the Cartan–Kähler theorem, which only works for real analytic differential systems, and the Cartan–Kuranishi prolongation theorem. See Further reading for details. The Newlander-Nirenberg theorem gives integrability conditions for an almost-complex structure.