Riesz–Thorin theorem: Difference between revisions

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In [[mathematics]], certain systems of [[partial differential equation]]s are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of [[differential form]]s. 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 system]]s, for example. A '''Pfaffian system''' is one specified by 1-forms alone, but the theory includes other types of example of '''differential system'''.
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Given a collection of differential 1-forms α<sub>''i''</sub>, ''i''=1,2, ..., ''k'' on an ''n''-dimensional manifold ''M'',  an '''integral manifold''' is a submanifold whose tangent space at every point ''p'' ∈ ''M'' is annihilated by each α<sub>''i''</sub>.
 
A '''maximal integral manifold''' is a 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 α<sub>''i''</sub> at every point ''p'' of ''N''. If in addition the α<sub>''i''</sub> are linearly independent, then ''N'' is (''n'' &minus; ''k'')-dimensional. Note that ''i'': ''N'' ⊂ ''M'' 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<sub>''i''</sub> 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 (differential topology)|Frobenius theorem]]. One version states that if the ideal <math>\mathcal I</math> algebraically generated by the collection of α<sub>''i''</sub> 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 system==
Not every Pfaffian system  is completely integrable in the Frobenius sense. For example, consider the following one-form on '''R'''<sup>3</sup> - (0,0,0)
 
:<math>\theta=x\,dy+y\,dz+z\,dx.</math>
 
If ''d''θ 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 '''R'''<sup>3</sup>.  Therefore, there are no two-dimensional leaves, and the system is not completely integrable.
 
On the other hand, the curve defined by
 
:<math> x =t, \quad y= c,  \qquad z = e^{-{t \over c}},  \quad t > 0 </math>
 
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<sup>''i''</sup>,  i.e., a collection of 1-forms forming a basis of the cotangent space at every point with <math>\langle\theta^i,\theta^j\rangle=\delta^{ij}</math> which are closed (dθ<sup>''i''</sup> = 0, i=1,2, ..., ''n'').  By the [[Poincaré lemma]], the θ<sup>''i''</sup> locally will have the form d''x<sup>i</sup>'' for some functions ''x<sup>i</sup>'' on the manifold, and thus provide an isometry of an open subset of M with an open subset of '''R'''<sup>''n''</sup>.  Such a manifold is called '''locally flat.'''
 
This problem reduces to a question on the [[frame bundle|coframe bundle]] of ''M''.  Suppose we had such a closed coframe
 
:<math>\Theta=(\theta^1,\dots,\theta^n)</math>.
 
If we had another coframe <math>\Phi=(\phi^1,\dots,\phi^n)</math>, 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
<math>d\omega+\omega\wedge\omega=0,</math> 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.
 
==Generalizations==
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 analysis|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, 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
 
[[Category:Partial differential equations]]
[[Category:Differential topology]]
[[Category:Differential systems]]

Latest revision as of 23:45, 2 January 2015

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