Integrability conditions for differential systems

From Exampleproblems

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

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.

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.

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.

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.

Integrability conditions for differential systems

From Exampleproblems

Contents

Example of a non-integrable system

Necessary and sufficient conditions

Examples

Generalizations

Further reading

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