Proof of Frobenius' Theorem: Directly Showing ##\omega \wedge d\omega = 0##

In summary, the conversation was about the proof of Frobenius' theorem, which is a theorem in differential geometry that states that if a certain condition is satisfied, then a one-form can be written in a specific form. The conversation included discussions about how this condition is locally and globally applied, and different interpretations of the theorem.
  • #36
cianfa72 said:
My concern is that Frobenius theorem (as stated here) involves immersed submanifolds and the Lemniscate curve is an immersed submanifold in ##\mathbb R^2##.
So what? The theorem does not say that any immersed submanifold must be a leaf of a foliation. Nor does it give any conditions that tell you whether an immersed submanifold is or is not a leaf of a foliation. So the theorem has nothing to say about your lemniscate curve.

The fact that the words "immersed submanifold" appear in your reference's statement of the theorem does not mean that the theorem must automaticallly decide all questions you have about immersed submanifolds.
 
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  • #37
cianfa72 said:
there is not any smooth vector field that restricted to the Lemniscate gives the tangent vector at each point along it.
Can you prove this or are you just waving your hands?
 
  • #38
martinbn said:
there isn't a smooth vector field that restricts to the tangent field along the curve.
Why do you think this? (Note, I'm not saying you're wrong, I'm just asking for a more explicit argument if you have one.)
 
  • #39
PeterDonis said:
Why do you think this? (Note, I'm not saying you're wrong, I'm just asking for a more explicit argument if you have one.)
Consider an open neighborhood of the origin ##(0,0)## in ##\mathbb R^2## and apply what @martinbn pointed out in his post #29.
 
  • #40
cianfa72 said:
Consider an open neighborhood of the origin ##(0,0)## in ##\mathbb R^2## and apply what @martinbn pointed out in his post #29.
Ah, got it.
 
  • #41
PeterDonis said:
The fact that the words "immersed submanifold" appear in your reference's statement of the theorem does not mean that the theorem must automaticallly decide all questions you have about immersed submanifolds.
Maybe I was unclear: my point was that it must exist at least an "immersed submanifold" (that however is not a regular embedded submanifold) which is an integral submanifold of some smooth vector field ##X## defined on the 'ambient' manifold. Otherwise the use of the word "immersed submanifold" in the theorem's statement does not make any sense.

The lemniscate curve was just an example of immersed submanifold that is not a regular embedded submanifold and we found that it is not actually a good example though.
 
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  • #42
cianfa72 said:
my point was that it must exist at least an "immersed submanifold" (that however is not a regular embedded submanifold) which is an integral submanifold of some smooth vector field ##X## defined on the 'ambient' manifold.
What "must" exist in that way? Why is this even an issue?

cianfa72 said:
Otherwise the use of the word "immersed submanifold" in the theorem's statement does not make any sense.
Yes, fine, what does that have to do with the lemniscate?

cianfa72 said:
The lemniscate curve was just an example of immersed submanifold that is not a regular embedded submanifold and we found that it is not actually a good example though.
Indeed. So why are we still discussing it?
 
  • #43
PeterDonis said:
Yes, fine, what does that have to do with the lemniscate?
Nothing, mine was just an example to help me in clarify the real content of the theorem.

Edit: Btw do you know an example of immersed submanifold (that is not however a regular embedded submanifold) that is a leaf of the foliation associated to some smooth vector field ?
 
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  • #44
Searching for it I found this interesting example of immersed submanifold (a 1D curve) that is not an embedding such that there is a smooth vector field ##X## that restricts to the tangent field along the curve.

The example involves a torus ##\mathbb R^2/ \mathbb Z^2## and a constant smooth vector field ##X## defined on the torus such that its integral curves are straight lines with slope an irrational number.
 
  • #45
cianfa72 said:
Searching for it I found this interesting example of immersed submanifold (a 1D curve) that is not an embedding such that there is a smooth vector field ##X## that restricts to the tangent field along the curve.

The example involves a torus ##\mathbb R^2/ \mathbb Z^2## and a constant smooth vector field ##X## defined on the torus such that its integral curves are straight lines with slope an irrational number.
A curve that turns and touches itself so that there is just one tangent line will be an example.
 
  • #46
martinbn said:
A curve that turns and touches itself so that there is just one tangent line will be an example.
But such a curve that turns and touches itself cannot be an immersed submanifold of the real line ##\mathbb R## (AFAIK the definition of immersed submanifold requires an injective map).
 
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  • #47
cianfa72 said:
But such a curve that turns and touches itself cannot be an immersed submanifold of the real line ##\mathbb R## (AFAIK the definition of immersed submanifold requires an injective map).
Injective only on tangent vectors, not the map itself.
 
  • #48
martinbn said:
Injective only on tangent vectors, not the map itself.
It depends on the definition used: for example some source requires also the injectivity of the map -- see for instance here at section 4.2.2 (pag 48).
 
  • #49
cianfa72 said:
It depends on the definition used: for example some source requires also the injectivity of the map -- see for instance here at section 4.2.2 (pag 48).
No, read it more carefully. Injective diferential, not the map. If the map is also injective then it is an embedding.
 
  • #50
martinbn said:
Injective diferential, not the map. If the map is also injective then it is an embedding.
See Definition 4.7 a) in section 4.2.2:
If ##f## is an injective immersion, then ##N## is called immersed smooth submanifold of M through ##f##.
 
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  • #51
martinbn said:
No, read it more carefully. Injective diferential, not the map. If the map is also injective then it is an embedding.
Not necessarily, the irrational curve on the torus is typically not considered an embedding.
 
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<h2>1. What is Frobenius' Theorem?</h2><p>Frobenius' Theorem is a mathematical theorem that states that if a differential form ##\omega## satisfies the condition ##\omega \wedge d\omega = 0##, then it is locally exact, meaning it can be written as the exterior derivative of another form.</p><h2>2. What does it mean for ##\omega \wedge d\omega = 0##?</h2><p>This condition means that the exterior product of the differential form ##\omega## with its exterior derivative ##d\omega## is equal to zero. In other words, the form ##\omega## is closed, or exact.</p><h2>3. How is Frobenius' Theorem used in mathematics?</h2><p>Frobenius' Theorem is used in various areas of mathematics, including differential geometry, differential equations, and algebraic geometry. It provides a useful tool for determining when a differential form is locally exact, which can help in solving various mathematical problems.</p><h2>4. Can Frobenius' Theorem be proved directly?</h2><p>Yes, Frobenius' Theorem can be proved directly by showing that the condition ##\omega \wedge d\omega = 0## implies that the form ##\omega## is locally exact. This proof involves using the Poincaré lemma and the concept of integrability of distributions.</p><h2>5. What are the applications of Frobenius' Theorem?</h2><p>Frobenius' Theorem has various applications in mathematics, including in the study of symplectic manifolds, Hamiltonian mechanics, and Lie groups. It also has applications in physics, particularly in the study of differential equations and conservation laws.</p>

1. What is Frobenius' Theorem?

Frobenius' Theorem is a mathematical theorem that states that if a differential form ##\omega## satisfies the condition ##\omega \wedge d\omega = 0##, then it is locally exact, meaning it can be written as the exterior derivative of another form.

2. What does it mean for ##\omega \wedge d\omega = 0##?

This condition means that the exterior product of the differential form ##\omega## with its exterior derivative ##d\omega## is equal to zero. In other words, the form ##\omega## is closed, or exact.

3. How is Frobenius' Theorem used in mathematics?

Frobenius' Theorem is used in various areas of mathematics, including differential geometry, differential equations, and algebraic geometry. It provides a useful tool for determining when a differential form is locally exact, which can help in solving various mathematical problems.

4. Can Frobenius' Theorem be proved directly?

Yes, Frobenius' Theorem can be proved directly by showing that the condition ##\omega \wedge d\omega = 0## implies that the form ##\omega## is locally exact. This proof involves using the Poincaré lemma and the concept of integrability of distributions.

5. What are the applications of Frobenius' Theorem?

Frobenius' Theorem has various applications in mathematics, including in the study of symplectic manifolds, Hamiltonian mechanics, and Lie groups. It also has applications in physics, particularly in the study of differential equations and conservation laws.

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