Integrability conditions for differential 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 ${\displaystyle \textstyle \alpha _{i},i=1,2,\dots ,k}$ on an ${\displaystyle \textstyle n}$-dimensional manifold ⁠${\displaystyle M}$⁠, an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point ${\displaystyle \textstyle p\in N}$ is annihilated by (the pullback of) each ⁠${\displaystyle \textstyle \alpha _{i}}$⁠.

A maximal integral manifold is an immersed (not necessarily embedded) submanifold

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

such that the kernel of the restriction map on forms

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

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

A Pfaffian system is said to be completely integrable if ${\displaystyle M}$ 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 ${\displaystyle \alpha _{i}}$ to guarantee that there will be integral submanifolds of sufficiently high dimension.

A Pfaffian system is specified by 1-forms. At each point ${\displaystyle x\in M}$, 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 ${\displaystyle T_{x}M}$. 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 ${\displaystyle M}$ 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 ${\displaystyle \mathbb {R} ^{3}}$, defined by the 1-form ${\displaystyle dz-ydx}$.

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 ${\displaystyle {\mathcal {I}}}$ algebraically generated by the collection of α_i inside the ring Ω(M) is differentially closed, in other words

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

then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.)

Given any regular foliation, we can simply take its differentials to obtain a 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.

Similar to integrable regular systems, a singular foliation produces an integrable singular system. For example, ${\displaystyle \mathbb {R} ^{3}}$ can be foliated into concentric circles with a singular point at the origin. This corresponds to an integrable singular system $${\displaystyle d(x^{2}+y^{2}+z^{2})=0\implies xdx+ydy+zdz=0}$$

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

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

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

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

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 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

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

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 general, a 1-form ${\displaystyle \theta }$ in a manifold of dimension ${\displaystyle 2n+1}$ is completely non-integrable iff ${\displaystyle \theta \wedge d\theta ^{n}\neq 0}$ everywhere. By a theorem of Pfaff, generalized by Darboux's theorem, there exists local coordinates in which it is of the form ${\displaystyle dz-\sum _{i=1}^{n}y_{i}dx_{i}}$. Such structures are contact structures.

Analogously, in an even-dimensional manifold, a 1-form ${\displaystyle \theta }$ in a manifold of dimension ${\displaystyle 2n+2}$ is completely non-integrable iff ${\displaystyle \theta \wedge d\theta ^{n}\neq 0}$ everywhere. Such structures are even-contact structures.

The 3-sphere ${\textstyle \mathbb {S} ^{3}}$ can be given a contact structure by considering it as the unit sphere in ${\textstyle \mathbb {C} ^{2}}$. The standard contact form on ${\textstyle \mathbb {S} ^{3}}$ is:$${\displaystyle \alpha ={\frac {1}{2}}\left(x_{1}dy_{1}-y_{1}dx_{1}+x_{2}dy_{2}-y_{2}dx_{2}\right)}$$where ${\textstyle \left(x_{1},y_{1},x_{2},y_{2}\right)}$ are coordinates on ${\textstyle \mathbb {R} ^{4}}$.

Some Pfaffian systems do not have a maximal integrable foliation, but are also not completely non-integrable.

For example, the standard contact structure on ${\displaystyle \mathbb {R} ^{3}}$, defined by the 1-form ${\displaystyle dx_{0}-x_{2}dx_{1}}$, is completely non-integrable, in the sense that any integral manifold of it can have only 1 dimension (these are called the Legendrian submanifolds). However, if we were to extend to ${\displaystyle \mathbb {R} ^{5}}$, then ${\displaystyle dx_{0}-x_{2}dx_{1}}$ can have an integral manifold of 3 dimensions. This is not the lowest dimension achievable, so it is neither completely integrable nor completely non-integrable, making it partially integrable.

In ${\displaystyle \mathbb {R} ^{5}}$, a completely integrable 1-form would have integral manifolds of 4 dimensions, and the standard contact structure ${\displaystyle dx_{0}-x_{2}dx_{1}-x_{4}dx_{3}}$ can only have integral manifolds of 2 dimensions, which makes it completely non-integrable.

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 ${\displaystyle \langle \theta ^{i},\theta ^{j}\rangle =\delta ^{ij}}$ 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

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

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

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

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

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

On the other hand,

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

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