Proving Existence of Vector Field X for 1-Form w on Smooth Manifold M

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Discussion Overview

The discussion revolves around the existence of a vector field \( X \) on a smooth manifold \( M \) such that a given 1-form \( w \) satisfies \( w(X) = f \), where \( f \) is a continuous function from \( M \) to \( \mathbb{R} \). The scope includes theoretical considerations regarding differential forms and vector fields on manifolds.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant initially poses the question of whether a vector field \( X \) exists such that \( w(X) = f \), indicating a desire for proof.
  • Another participant expresses confusion regarding the formulation, stating that a 1-form acting on a vector field should yield a real number, not a function \( f \).
  • A clarification is made that \( w \) is a 1-form and \( f \) is a continuous function, restating the question about the existence of \( X \) in a neighborhood of a point \( p \) in \( M \).
  • One participant asserts that there are generally infinitely many vector fields \( X \) that can satisfy the equation, depending on the local chart around \( p \).
  • Another participant notes that it is also possible for there to be no such vector field if \( w \) is the zero 1-form and \( f \) is non-zero.
  • Further clarification is provided regarding the linearity of operations on each fiber, suggesting that if one vector field \( X \) satisfies \( w(X) = f \), then any vector field of the form \( X + Y \) (where \( Y \) satisfies \( w(Y) = 0 \)) will also satisfy \( w(X + Y) = f \).
  • One participant questions the formulation of the original question, asking if there is a specific application or context that prompted the inquiry.

Areas of Agreement / Disagreement

Participants express differing views on the existence of the vector field \( X \). While some suggest that infinitely many solutions exist, others point out the possibility of no solutions under certain conditions. The discussion remains unresolved regarding the optimal formulation of the question and the motivations behind it.

Contextual Notes

The discussion highlights the dependence on local charts and the properties of 1-forms and vector fields. There is an acknowledgment of potential ambiguities in the original question and the implications of the zero 1-form.

daishin
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Let w be a 1-form on smooth manifold M. Then is there a vector field X such that locally w(X)=f where f:M-->R continuous?
How can I prove it?
Thanks.
 
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i'm a bit confused, your 1-form acting on the vector field should yield a real number (not f)..
 
Sorry what I meant was:
Let w be a 1-form on smooth manifold M. Let f be a continuous function from M to R.
Then is there a vector field X on M such that w(X)=f in some neighborhood of p in M?
 
Last edited:
Yes. Take a local chart around p and write the problem. You will see that there are in general infinitely many X satisfying this.
 
timur said:
Yes. Take a local chart around p and write the problem. You will see that there are in general infinitely many X satisfying this.

Or none. Let w=0(the 0 1-form). Let f be a non-zero function.
 
quetzalcoatl9 said:
i'm a bit confused, your 1-form acting on the vector field should yield a real number (not f)..

a differential 1-form on a manifold acting on a vector field on a manifold yields a function.
 
daishin said:
Sorry what I meant was:
Let w be a 1-form on smooth manifold M. Let f be a continuous function from M to R.
Then is there a vector field X on M such that w(X)=f in some neighborhood of p in M?

The operations are linear on each fiber. So, if you solve w(Y)=0 and find one X such that w(X)=f, then X+Y is such that w(X+Y)=f.

The question is not optimally formulated, and it is a little unclear why you are asking this question. Do you have an application in mind? Are you reading a proof in a book or trying to do a problem?
 

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