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

Given a collection of differential 1-forms αi, i=1,2, ..., k on an n-dimensional manifold M, an integral manifold is a submanifold whose tangent space at every point pM is annihilated by each αi.

A maximal integral manifold is a 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 αi at every point p of N. If in addition the αi are linearly independent, then N is (nk)-dimensional. Note that i: NM need not be an embedded submanifold.

A Pfaffian system is said to be completely integrable if N 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 αi to guarantee that there will be integral submanifolds of sufficiently high dimension.

Necessary and sufficient conditions[edit]

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 αi inside the ring Ω(M) is differentially closed, in other words

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

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

Example of a non-integrable system[edit]

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

\theta=x\,dy+y\,dz+z\,dx.

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, the curve defined by

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

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[edit]

In Riemannian geometry, we may consider the problem of finding an orthogonal coframe θi, i.e., a 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 (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

\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

\Phi=M\Theta

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

d\Phi=\omega\wedge\Phi

On the other hand,


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

But \omega=(dM)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[edit]

Many generalizations exist to integrability conditions on differential systems which 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.

Further reading[edit]

  • 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