Inmathematics, certain systems ofpartial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system ofdifferential forms. The idea is to take advantage of the way a differential formrestricts to asubmanifold, and the fact that this restriction is compatible with theexterior derivative. This is one possible approach to certainover-determined systems, for example, includingLax pairs ofintegrable systems. APfaffian system is specified by1-forms alone, but the theory includes other types of example ofdifferential system. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold (which one sets equal to 0 to findsolutions to the system).

Given a collection of differential 1-forms${\displaystyle {\alpha }_i,i=1,2,\dots ,k}$ on an${\displaystyle n}$-dimensional manifold⁠${\displaystyle M}$⁠, anintegral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point${\displaystyle p\in N}$ is annihilated by (the pullback of) each⁠${\displaystyle {\alpha }_i}$⁠.

Amaximal integral manifold is an immersed (not necessarily embedded) submanifold

${\displaystyle i:N\subset M}$

such that the kernel of the restriction map on forms

${\displaystyle i^{\ast }:{\mathrm{\Omega }}_p^1(M)\to {\mathrm{\Omega }}_p^1(N)}$

is spanned by the${\displaystyle {\alpha }_i}$ at every point${\displaystyle p}$ of⁠${\displaystyle N}$⁠. If in addition the${\displaystyle {\alpha }_i}$ are linearly independent, then${\displaystyle N}$ is (⁠${\displaystyle n-k}$⁠)-dimensional.

A Pfaffian system is said to becompletely integrable if${\displaystyle M}$ admits afoliation by maximal integral manifolds. (Note that the foliation need not beregular; i.e. the leaves of the foliation might not be embedded submanifolds.)

Anintegrability condition is a condition on the${\displaystyle {\alpha }_i}$ to guarantee that there will be integral submanifolds of sufficiently high dimension.

The necessary and sufficient conditions forcomplete integrability of a Pfaffian system are given by theFrobenius theorem. One version states that if the ideal${\displaystyle \mathcal{I}}$ algebraically generated by the collection ofα_i inside the ring Ω(M) is differentially closed, in other words

${\displaystyle d\mathcal{I}\subset \mathcal{I},}$

then the system admits afoliation 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-formonR³ ∖ (0,0,0):

${\displaystyle \theta =zdx+xdy+ydz.}$

Ifdθ were in the ideal generated byθ we would have, by the skewness of the wedge product

${\displaystyle \theta \wedge d\theta =0.}$

But a direct calculation gives

${\displaystyle \theta \wedge d\theta =(x+y+z)dx\wedge dy\wedge dz,}$

which is a nonzero multiple of the standard volume form onR³. Therefore, there are no two-dimensional leaves, and the system is not completely integrable.

On the other hand, for the curve defined by

${\displaystyle x=t,y=c,z=e^{-t/c},t>0}$

thenθ defined as above is 0, and hence the curve is easily verified to be a solution (i.e. anintegral curve) for the above Pfaffian system for any nonzero constantc.

Inpseudo-Riemannian geometry, we may consider the problem of finding an orthogonalcoframeθⁱ, i.e., a collection of 1-forms that form a basis of the cotangent space at every point with${\displaystyle ⟨{\theta }^i,{\theta }^j⟩={\delta }^{ij}}$ that are closed (dθⁱ = 0,i = 1, 2, ...,n). By thePoincaré lemma, theθⁱ locally will have the formdxⁱ for some functionsxⁱ on the manifold, and thus provide an isometry of an open subset ofM with an open subset ofRⁿ. Such a manifold is calledlocally flat.

This problem reduces to a question on thecoframe bundle ofM. Suppose we had such a closed coframe

${\displaystyle \mathrm{\Theta }=({\theta }^1,\dots ,{\theta }^n).}$

If we had another coframe⁠${\displaystyle \mathrm{\Phi }=({\varphi }^1,\dots ,{\varphi }^n)}$⁠, then the two coframes would be related by an orthogonal transformation

${\displaystyle \mathrm{\Phi }=M\mathrm{\Theta }}$

If the connection 1-form isω, then we have

${\displaystyle d\mathrm{\Phi }=\omega \wedge \mathrm{\Phi }}$

On the other hand,

But${\displaystyle \omega =(dM)M^{-1}}$ is theMaurer–Cartan form for theorthogonal group. Therefore, it obeys the structural equation⁠${\displaystyle d\omega +\omega \wedge \omega =0}$⁠, and this is just thecurvature ofM:${\displaystyle \mathrm{\Omega }=d\omega +\omega \wedge \omega =0.}$After an application of the Frobenius theorem, one concludes that a manifoldM is locally flat if and only if its curvature vanishes.

Many generalizations exist to integrability conditions on differential systems that are not necessarily generated by one-forms. The most famous of these are theCartan–Kähler theorem, which only works forreal analytic differential systems, and theCartan–Kuranishi prolongation theorem. See§ Further reading for details. TheNewlander–Nirenberg theorem gives integrability conditions for an almost-complex structure.

• Bryant, Chern, Gardner, Goldschmidt, Griffiths,Exterior Differential Systems, Mathematical Sciences Research Institute Publications, Springer-Verlag,ISBN 0-387-97411-3
• Olver, P.,Equivalence, Invariants, and Symmetry, Cambridge,ISBN 0-521-47811-1
• Ivey, T., Landsberg, J.M.,Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, American Mathematical Society,ISBN 0-8218-3375-8
• Dunajski, M.,Solitons, Instantons and Twistors, Oxford University Press,ISBN 978-0-19-857063-9

• Partial differential equations
• Differential topology
• Differential systems