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

[daishin]

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.

 

[quetzalcoatl9]

i'm a bit confused, your 1-form acting on the vector field should yield a real number (not f)..

 

[daishin]

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?

 

[timur]

Yes. Take a local chart around p and write the problem. You will see that there are in general infinitely many X satisfying this.

 

[Reverie]

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.

 

[Reverie]

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.

 

[Reverie]

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?