Talk:Frobenius theorem (differential topology)

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I'd like to refine the second paragraph of the Introduction: It should be something like "A smooth vector field on a smooth manifold M gives rise to a one-parameter family of diffeomorphisms, which can be integrated locally to give integral curves through a point, and this can be done globally if M is compact." Any thoughts?

Does anyone know of any research on the validity of the result in an infinite dimensional setting?

There is a standard version for Banach manifolds, but I would need to look up the precise statement. It is likely that there is also a version for tame Frèchet manifolds as well, but wikipedia lacks an article describing that category and I'm gradually getting around to it. Silly rabbit 03:29, 18 June 2006 (UTC)

[edit] Todo list

• Work in the fact that this is a local theorem in the intro.
• Check the Deahna paper. I'm fairly sure Clebsch proved the PDE version of this theorem, but I'm not certain about Deahna's contribution.
• In intro, figure out a way to say how the level sets (i.e., integral manifolds) are related to initial value problems.
• Relate the vector field formulation to the PDE version.
• Rework the differential form version.
• Infinite dimensional cases.