Talk:Frobenius theorem (differential topology)
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I'm just now taking an introductary differential geometry course, which stated this theorem in a form really different from anything here: if X_1,...,X_n are n vector fields on a smooth n-manifold, then there's a local diffeomorphism F from a neighbourhood of a point p in M such that F(X_i) = d/(dx_i) iff the vector fields are nonzero at p and satisfy [X_i,X_j]_p = 0 when i \neq j. Maybe the article could clarify how/why this statement is equivalent to the given formulations?
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.
