Integrability conditions for differential systems

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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. A Pfaffian system is one specified by 1-forms alone, but the theory includes other types of example of differential system.

Given a collection of differential 1-forms $\alpha_i, i=1,2,\dots,k$ on an n-dimensional manifold M, an integral submanifold is an embedding

$i:N\subset M$

of a submanifold N into 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 $α i$ at every point p of N. If in addition the $α i$ are linearly independent, then N is (n − k)-dimensional.

An integrability condition is a condition on the $α i$ to guarantee that there will be an integral submanifold.

1 Example of a non-integrable system
2 Necessary and sufficient conditions
3 Examples
4 Generalizations
5 Further reading

[edit] Example of a non-integrable system

Not every such differential system has integral manifolds, however. For example, consider the following one-form on the standard simplex $S=\{(x,y,z)|x+y+z<1\}\subset\mathbb R^3$ :

θ = x d y + y d z + z d x

Suppose that N is an integral submanifold for $θ$ , so that $i * θ = 0$ . In particular, $i * d θ = d i * θ = 0.$ So $d θ$ is also in the kernel of $i *$ , which means that we must have $d\theta=\alpha\wedge\theta$ for some 1-form $α$ on M. On the other hand, by the skewness of the wedge product, this implies that

$\theta\wedge d\theta=0.$

But a direct calculation verifies that

$\theta\wedge d\theta=(x+y+z)dx\wedge dy\wedge dz$

which is a nonzero multiple of the standard volume on the simplex S, and so is never zero.

[edit] Necessary and sufficient conditions

The necessary and sufficient conditions for integrability of a system generated by 1-forms are supplied by the Frobenius theorem. One form states that if the ideal $\mathcal I$ algebraically generated by the collection of $α i$ inside the ring $Ω(M)$ is differentially closed $d{\mathcal I}\subset {\mathcal I}$ , then the system admits an integral manifold.

[edit] Examples

In Riemannian geometry, we may consider the problem of finding an orthogonal coframe $θ i$ (i.e., collection of 1-forms forming a basis of the cotangent space at every point with $\langle\theta^i,\theta^j\rangle=\delta^{ij}$ ) which are closed $d\theta^i=0, i=1,2,\dots,n$ . By the Poincaré lemma, the $θ i$ locally will have the form $d x i$ for some functions $x i$ on the manifold, and thus provide an isometry of an open subset of M with an open subset of $\mathbb R^n$ . 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 ${\Phi}=(\phi^1,\dots,\phi^n)$ , then the two coframes would be related by an orthogonal transformation

Φ = M Θ

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

$d\Phi=\omega\wedge\Phi$

On the other hand,

$d\Phi\,$	$=(dM)\wedge\Theta+M\wedge d\Theta$
	$=(dM)\wedge\Theta$
	$=(dM)M^{-1}\wedge\Phi.$

But $ω = (d M) M - 1$ is the Maurer-Cartan form for the orthogonal group. Therefore it obeys the structural equation $d\omega+\omega\wedge\omega=0,$ 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.

[edit] Generalizations

Many generalizations exist to integrability conditions on differential systems which are not necessarily generated by one-forms. The most famous of which are the Cartan-Kähler theorem, which only works for real analytic differential systems, and the Cartan-Kuranishi prolongation theorem. See Further reading for details.

[edit] Further reading

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

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Categories: Partial differential equations | Differential topology