 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 overdetermined systems, for example. A Pfaffian system is one specified by 1forms alone, but the theory includes other types of example of differential system.
Given a collection of differential 1forms on an ndimensional manifold M, an integral manifold is a submanifold whose tangent space at every point is annihilated by each α_{i}.
A maximal integral manifold is a submanifold
such that the kernel of the restriction map on forms
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. Note that 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 α_{i} to guarantee that there will be integral submanifolds of sufficiently high dimension.
Contents
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 α_{i} 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 nonintegrable system
Not every Pfaffian system is completely integrable in the Frobenius sense. For example, consider the following oneform on R^{3} − (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 R^{3}. Therefore, there are no twodimensional leaves, and the system is not completely integrable.
On the other hand, the curve defined by
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 θ^{i} (i.e., collection of 1forms forming a basis of the cotangent space at every point with ) which are closed . By the Poincaré lemma, the θ^{i} locally will have the form dx^{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 . 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
 Φ = MΘ
If the connection 1form is ω, then we have
On the other hand,
But ω = (dM)M ^{− 1} 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.
Generalizations
Many generalizations exist to integrability conditions on differential systems which are not necessarily generated by oneforms. The most famous of which are the CartanKähler theorem, which only works for real analytic differential systems, and the Cartan–Kuranishi prolongation theorem. See Further reading for details.
Further reading
 Bryant, Chern, Gardner, Goldschmidt, Griffiths, Exterior Differential Systems, Mathematical Sciences Research Institute Publications, SpringerVerlag, ISBN 0387974113
 Olver, P., Equivalence, Invariants, and Symmetry, Cambridge, ISBN 0521478111
 Ivey, T., Landsberg, J.M., Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, American Mathematical Society, ISBN 0821833758
Categories: Partial differential equations
 Differential topology
 Differential systems
Wikimedia Foundation. 2010.
Look at other dictionaries:
Frobenius theorem (differential topology) — In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first order homogeneous linear partial differential equations. In modern geometric terms … Wikipedia
List of differential geometry topics — This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. Contents 1 Differential geometry of curves and surfaces 1.1 Differential geometry of curves 1.2 Differential… … Wikipedia
Distribution (differential geometry) — For other meanings, see Distribution (disambiguation). In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties. Distributions are used to build up… … Wikipedia
Spectral theory of ordinary differential equations — In mathematics, the spectral theory of ordinary differential equations is concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl… … Wikipedia
Integrable system — In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems. In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In… … Wikipedia
Connexion affine — Une connexion affine sur la sphère fait rouler le plan affine tangent d un point à un autre. Dans ce déplacement, le point de contact trace une courbe du plan : le développement. En mathématiques, et plus précisément en géométrie… … Wikipédia en Français
Élie Cartan — Infobox Person name = Élie Joseph Cartan image size = 200px caption = Professor Élie Joseph Cartan birth date = birth date186949 birth place = Dolomieu, Savoie, France death date = death date and age195156186949 death place = Paris,… … Wikipedia
List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… … Wikipedia
Connexion de Ehresmann — En géométrie différentielle, une connexion de Ehresmann (d après le mathématicien français Charles Ehresmann qui a le premier formalisé ce concept) est une version de la notion de connexion qui est définie sur des fibrés. En particulier, elle… … Wikipédia en Français
Cartan–Kähler theorem — In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I . It is named for Élie Cartan and Erich Kähler.It is not true that… … Wikipedia