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. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold (which one sets equal to 0 to find solutions to the system).
Given a collection of differential 1-forms <math>\textstyle\alpha_i, i=1,2,\dots, k</math> on an <math>\textstyle n</math>-dimensional manifold Template:Tmath, an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point <math>\textstyle p\in N</math> is annihilated by (the pullback of) each Template:Tmath.
A maximal integral manifold is an immersed (not necessarily embedded) submanifold
- <math>i:N\subset M</math>
such that the kernel of the restriction map on forms
- <math>i^*:\Omega_p^1(M)\rightarrow \Omega_p^1(N)</math>
is spanned by the <math>\textstyle \alpha_i</math> at every point <math>p</math> of Template:Tmath. If in addition the <math>\textstyle \alpha_i</math> are linearly independent, then <math>N</math> is (Template:Tmath)-dimensional.
A Pfaffian system is said to be completely integrable if <math>M</math> 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 <math>\alpha_i</math> to guarantee that there will be integral submanifolds of sufficiently high dimension.
Necessary and sufficient conditionsEdit
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 <math>\mathcal I</math> algebraically generated by the collection of αi inside the ring Ω(M) is differentially closed, in other words
- <math>d{\mathcal I}\subset {\mathcal I},</math>
then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.)
Example of a non-integrable systemEdit
Not every Pfaffian system is completely integrable in the Frobenius sense. For example, consider the following one-form Template:Nowrap:
- <math>\theta=z\,dx +x\,dy+y\,dz.</math>
If dθ were in the ideal generated by θ we would have, by the skewness of the wedge product
- <math>\theta\wedge d\theta=0.</math>
But a direct calculation gives
- <math>\theta\wedge d\theta=(x+y+z)\,dx\wedge dy\wedge dz ,</math>
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
- <math> x = t, \quad y = c, \quad z = e^{-t/c}, \qquad t > 0 </math>
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 applicationsEdit
In pseudo-Riemannian geometry, we may consider the problem of finding an orthogonal coframe θi, i.e., a collection of 1-forms that form a basis of the cotangent space at every point with <math>\langle\theta^i,\theta^j\rangle=\delta^{ij}</math> that are closed (dθi = 0, Template:Nowrap). 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
- <math>\Theta=(\theta^1,\dots,\theta^n).</math>
If we had another coframe Template:Tmath, then the two coframes would be related by an orthogonal transformation
- <math>\Phi=M\Theta</math>
If the connection 1-form is ω, then we have
- <math>d\Phi=\omega\wedge\Phi</math>
On the other hand,
- <math>
\begin{align} d\Phi & = (dM)\wedge\Theta+M\wedge d\Theta \\ & =(dM)\wedge\Theta \\ & =(dM)M^{-1}\wedge\Phi. \end{align} </math>
But <math>\omega=(dM)M^{-1}</math> is the Maurer–Cartan form for the orthogonal group. Therefore, it obeys the structural equation Template:Tmath, and this is just the curvature of M: <math>\Omega=d\omega+\omega\wedge\omega=0.</math> After an application of the Frobenius theorem, one concludes that a manifold M is locally flat if and only if its curvature vanishes.
GeneralizationsEdit
Many generalizations exist to integrability conditions on differential systems that 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 Template:Slink for details. The Newlander–Nirenberg theorem gives integrability conditions for an almost-complex structure.
Further readingEdit
- Bryant, Chern, Gardner, Goldschmidt, Griffiths, Exterior Differential Systems, Mathematical Sciences Research Institute Publications, Springer-Verlag, Template:ISBN
- Olver, P., Equivalence, Invariants, and Symmetry, Cambridge, Template:ISBN
- Ivey, T., Landsberg, J.M., Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, American Mathematical Society, Template:ISBN
- Dunajski, M., Solitons, Instantons and Twistors, Oxford University Press, Template:ISBN