integrability condition

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 $\textstyle\alpha_i, i=1,2,\dots, k$ on an $\textstyle n$-dimensional manifold , an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point $\textstyle p\in N$ is annihilated by (the pullback of) each .

A maximal integral manifold is an immersed (not necessarily embedded) submanifold
: $i:N\subset M$
such that the kernel of the restriction map on forms
: $i^*:\Omega_p^1(M)\rightarrow \Omega_p^1(N)$
is spanned by the $\textstyle \alpha_i$ at every point $p$ of . If in addition the $\textstyle \alpha_i$ are linearly independent, then $N$ is ()-dimensional.

A Pfaffian system is said to be completely integrable if $M$ 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 $\alpha_i$ to guarantee that there will be integral submanifolds of sufficiently high dimension.

Necessary and sufficient conditions

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 $\mathcal I$ algebraically generated by the collection of ''α''_''i'' inside the ring Ω(''M'') is differentially closed, in other words
: $d\subset ,$
then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.)

Example of a non-integrable system

Not every Pfaffian system is completely integrable in the Frobenius sense. For example, consider the following one-form :
: $\theta=z\,dx +x\,dy+y\,dz.$

If ''dθ'' were in the ideal generated by ''θ'' we would have, by the skewness of the wedge product
: $\theta\wedge d\theta=0.$

But a direct calculation gives
: $\theta\wedge d\theta=(x+y+z)\,dx\wedge dy\wedge dz ,$
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
: $x = t, \quad y = c, \quad z = e^, \qquad t > 0$
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''.

Examples of applications

In pseudo-Riemannian geometry, we may consider the problem of finding an orthogonal coframe ''θ''^''i'', i.e., a collection of 1-forms that form a basis of the cotangent space at every point with $\langle\theta^i,\theta^j\rangle=\delta^$ that are closed (''dθ''^''i'' = 0, ). By the Poincaré lemma, the ''θ''^''i'' 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 R^''n''. 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
: $\Theta=(\theta^1,\dots,\theta^n).$

If we had another coframe , then the two coframes would be related by an orthogonal transformation
: $\Phi=M\Theta$

If the connection 1-form is ''ω'', then we have
: $d\Phi=\omega\wedge\Phi$

On the other hand,
: $\begin d\Phi & = (dM)\wedge\Theta+M\wedge d\Theta \\ & =(dM)\wedge\Theta \\ & =(dM)M^\wedge\Phi. \end$

But $\omega=(dM)M^$ is the Maurer–Cartan form for the orthogonal group. Therefore, it obeys the structural equation
, and this is just the curvature of ''M'': $\Omega=d\omega+\omega\wedge\omega=0.$
After an application of the Frobenius theorem, one concludes that a manifold ''M'' is locally flat if and only if its curvature vanishes.

Generalizations

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 ' for details. The Newlander–Nirenberg theorem gives integrability conditions for an almost-complex structure.

Further reading

* Bryant, Chern, Gardner, Goldschmidt, Griffiths, ''Exterior Differential Systems'', Mathematical Sciences Research Institute Publications, Springer-Verlag,
* Olver, P., ''Equivalence, Invariants, and Symmetry'', Cambridge,
* Ivey, T., Landsberg, J.M., ''Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems'', American Mathematical Society,
* Dunajski, M., ''Solitons, Instantons and Twistors'', Oxford University Press, {{ISBN, 978-0-19-857063-9

Partial differential equations
Differential topology
Differential systems