Integrability conditions for differential systems
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.
Given a collection of differential 1-forms on an -dimensional manifold , an integral manifold is a submanifold whose tangent space at every point is annihilated by each .
A maximal integral manifold is a submanifold
such that the kernel of the restriction map on forms
is spanned by the at every point of . If in addition the are linearly independent, then is ()-dimensional. Note that need not be an embedded submanifold.
A Pfaffian system is said to be completely integrable if 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 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 algebraically generated by the collection of αi inside the ring Ω(M) is differentially closed, in other words
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 on R3 − (0,0,0):
If dθ were in the ideal generated by θ we would have, by the skewness of the wedge product
But a direct calculation gives
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
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 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 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
If we had another coframe , then the two coframes would be related by an orthogonal transformation
If the connection 1-form is ω, then we have
On the other hand,
But is the Maurer–Cartan form for the orthogonal group. Therefore, it obeys the structural equation and this is just the curvature of M: 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 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.
- 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
- Dunajski, M., Solitons, Instantons and Twistors, Oxford University Press, ISBN 978-0-19-857063-9