Talk:Frobenius theorem (differential topology)

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)

As a matter of fact, in the holomorphic version there is an error. It is important that the theorem proves also that: det(f^i_j)(p)\neq 0. —Preceding unsigned comment added by 157.82.237.53 (talk) 06:54, 17 March 2009 (UTC)

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.


 * Make the article more user-friendly by including appropriate pictures. The related article on Foliations has many nice pictures, some of which might well be appropriate here. Ishboyfay (talk) 18:09, 13 March 2022 (UTC)

over/under-determined
The first line says "solutions of an overdetermined system of first-order homogeneous linear partial differential equations." but then we see r<n equations in n variables, which is under-determined. Billlion (talk) 09:44, 27 November 2008 (UTC)


 * Originally the article said underdetermined as well, but someone (who I strongly suspect was the integrable systems expert User:R Physicist) changed it to overdetermined. I think most sources probably do not label it one or the other, and avoid a discussion of the over/under issue altogether.  However, the system has characteristics of both kinds of systems.  On the one hand, it is superficially like an underdetermined system because the number of unknowns exceeds the number of equations.  However, it is like an overdetermined system because unless a certain condition holds, the system fails to have solutions.  I'm not sure where that leaves the article.  R Physicist seemed to feel that it was important to mention it, otherwise I would suggest removing it altogether.   siℓℓy rabbit  (  talk  ) 12:59, 27 November 2008 (UTC)

regular foliation?
I was wondering what the role and definition is for the word "regular" in the article. Orthografer (talk) 21:47, 23 September 2009 (UTC)

A regular foliation is such, that the leafs are of const rank. Right? — Preceding unsigned comment added by 92.226.34.126 (talk) 20:04, 11 October 2011 (UTC)

Incomprehensible
Just a quick comment, this article is incomprehensible to anybody who does not already understand the subject. The article is not written like an encyclopedia article, but like a chapter from a math textbook (and a non-introductory text at that).

Summary: the article needs an introduction for non-experts. Skepticalgiraffe (talk) 15:00, 10 December 2018 (UTC)


 * A few pictures would go a long way toward making the article more comprehensible. Ishboyfay (talk) 18:09, 13 March 2022 (UTC)