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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 α 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 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 Ω p 1 ( M ) → → Ω Ω p 1 ( N ) {\displaystyle i^{*}:\Omega _{p}^{1}(M)\rightarrow \Omega _{p}^{1}(N)}

is spanned by the α α i {\displaystyle \textstyle \alpha _{i}} at every point p {\displaystyle p} of ⁠ N {\displaystyle N} ⁠. If in addition the α 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 α i {\displaystyle \alpha _{i}} to guarantee that there will be integral submanifolds of sufficiently high dimension.

• Necessary and sufficient conditions

• Example of a non-integrable system

• Examples of applications

• Generalizations

• Further reading

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.)

Not every Pfaffian system is completely integrable in the Frobenius sense. For example, consider the following one-form on R3 ∖ (0,0,0):

θ θ = z d x + x d y + y d z . {\displaystyle \theta =z\,dx+x\,dy+y\,dz.}

If dθ were in the ideal generated by θ we would have, by the skewness of the wedge product

θ θ ∧ ∧ d θ θ = 0. {\displaystyle \theta \wedge d\theta =0.}

But a direct calculation gives

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

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 {\displaystyle x=t,\quad y=c,\quad z=e^{-t/c},\qquad 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 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 ⟨ ⟨ θ θ i , θ θ j ⟩ ⟩ = δ δ i j {\displaystyle \langle \theta ^{i},\theta ^{j}\rangle =\delta ^{ij}} that are closed (dθi = 0, i = 1, 2, ..., n). 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.

This problem reduces to a question on the coframe bundle of M. Suppose we had such a closed coframe

Θ Θ = ( θ θ 1 , … … , θ θ n ) . {\displaystyle \Theta =(\theta ^{1},\dots ,\theta ^{n}).}

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

Φ Φ = M Θ Θ {\displaystyle \Phi =M\Theta }

If the connection 1-form is ω, then we have

d Φ Φ = ω ω ∧ ∧ Φ Φ {\displaystyle d\Phi =\omega \wedge \Phi }

On the other hand,

d Φ Φ = ( d M ) ∧ ∧ Θ Θ + M ∧ ∧ d Θ Θ = ( d M ) ∧ ∧ Θ Θ = ( d M ) M − − 1 ∧ ∧ Φ Φ . {\displaystyle {\begin{aligned}d\Phi &=(dM)\wedge \Theta +M\wedge d\Theta \\&=(dM)\wedge \Theta \\&=(dM)M^{-1}\wedge \Phi .\end{aligned}}}

But ω ω = ( d M ) M − − 1 {\displaystyle \omega =(dM)M^{-1}} is the Maurer–Cartan form for the orthogonal group. Therefore, it obeys the structural equation ⁠ d ω ω + ω ω ∧ ∧ ω ω = 0 {\displaystyle d\omega +\omega \wedge \omega =0} ⁠, and this is just the curvature of M: Ω Ω = d ω ω + ω ω ∧ ∧ ω ω = 0. {\displaystyle \Omega =d\omega +\omega \wedge \omega =0.} 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 that 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.

• 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