Frobenius theorem for differential one forms

[cianfa72]

TL;DR Summary

Frobenius theorem for differential one forms - equivalence of conditions

Hi, starting from this old PF thread I've some doubts about the Frobenius condition for a differential 1-form ##\omega##, namely that ##d\omega = \omega \wedge \alpha## is actually equivalent to the existence of smooth maps ##f## and ##g## such that ##\omega = fdg##.

I found this About Frobenius's theorem for differential forms where the OP asks for an "algebraic" proof of the equivalence ##d\omega = \omega \wedge \alpha \Leftrightarrow\omega = f dg##. The implication ##\Leftarrow## is algebraically clear, just take ##\alpha = - df/f##.

The other implication ##\Rightarrow## seems to be, instead, not algebraically straightforward (the fourth comment there shows a counterexample regarding the fact that the OP's proposal algebraic proof does not work).

Is basically this latter implication the actual content of Frobenius's theorem ? Thanks.

 

[cianfa72]

Said in other terms: in the simple case of just 1-form ##\omega## defined on a smooth manifold ##M##, the Frobenius condition ##\omega \wedge d\omega= 0## (that is equivalent to the existence of a 1-form ##\alpha## such that ##d\omega = \omega \wedge \alpha##) tell us that the ##n-1## dimensional smooth distribution is integrable (i.e. there exist an integral immersed submanifold of ##M## for each point ##p \in M##).

The above does mean there exist a complete foliation of ##M## via such maximal connected immersed submanifolds (see also here Lec11).

My point is that such leaves of the foliation can be always given as level sets of a smooth map ##t## defined on the manifold ##M##. Then ##\omega## can be always "recovered" from such a map ##t## using another smooth map ##f## as ##\omega=fdt##.

Does it make sense ?

 

[martinbn]

Yes, it is Frobenius theorem.

 

[cianfa72]

I checked also Lee's book Theorem 19.12 (Global Frobenius theorem): the foliation via maximal connected integral submanifolds does exist. Therefore, I believe, in case of an ##n-1## smooth involutive distribution assigned as the kernel of a 1-form ##\omega##, such submanifolds can be given as the level sets of a smooth function/map ##t## defined on the entire manifold ##M##. Hence it seems to me ##\omega = fdt## for smooth functions/maps ##f## and ##t## globally defined on ##M##.

 

[graphking]

can't see how to use Frobenius theorem, I think this is just a hard exercise. you can search in John lee, gtm218 for the true meaning of Frobenius theorem

 

[cianfa72]

you can search in John lee, gtm218 for the true meaning of Frobenius theorem

Yes, I've seen it on Jon Lee's Introduction on smooth manifolds.

 

[graphking]

Yes, I've seen it on Jon Lee's Introduction on smooth manifolds.

so how to use? can't see they are relevant