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.
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).
, an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point
A maximal integral manifold is an immersed (not necessarily embedded) submanifold such that the kernel of the restriction map on forms is spanned by the
A Pfaffian system is said to be completely integrable if
admits a foliation by maximal integral manifolds.
to guarantee that there will be integral submanifolds of sufficiently high dimension.
The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius theorem.
algebraically generated by the collection of αi inside the ring Ω(M) is differentially closed, in other words then the system admits a foliation by maximal integral manifolds.
Not every Pfaffian system is completely integrable in the Frobenius sense.
For example, consider the following one-form on R3 ∖ (0,0,0): If dθ were in the ideal generated by θ we would have, by the skewness of the wedge product But a direct calculation gives 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 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 pseudo-Riemannian geometry, we may consider the problem of finding an orthogonal coframe θi, i.e., a collection of 1-forms that form a basis of the cotangent space at every point with
By the Poincaré lemma, the θi locally will have the form dxi for some functions xi on the manifold, and thus provide an isometry of an open subset of M with an open subset of Rn.
Such a manifold is called locally flat.
, then the two coframes would be related by an orthogonal transformation If the connection 1-form is ω, then we have On the other hand, But
After an application of the Frobenius theorem, one concludes that a manifold M is locally flat if and only if its curvature vanishes.
Many generalizations exist to integrability conditions on differential systems thar are not necessarily generated by one-forms.
The Newlander–Nirenberg theorem gives integrability conditions for an almost-complex structure.