Smooth covector field on S^2

Different · #1 $Old$ April 2nd, 2009, 09:18 AM

Is there a smooth covector field on ${\mathbb S}^2$ that is exact and vanishes at exactly one point?

I think the answer is no...
in case of exact field we have $w = df$ for some smooth function on ${\mathbb S}^2$ ... it vanishes at some point $p$ when partial derivatives of $f$ at p are equal to zero (in some chart containing $p$ ).

..is it correct to say that if we consider stereographic coordinates on ${\mathbb S}^2$ then for any point $p$ where $w=df$ vanishes it will also vanish at point $-p$ ??

Rebesques · #2 $Old$ June 5th, 2009, 04:50 PM

Quote:

Originally Posted by Different $View Post$

Is there a smooth covector field on ${\mathbb S}^2$ that is exact and vanishes at exactly one point?

I think the answer is no...
in case of exact field we have $w = df$ for some smooth function on ${\mathbb S}^2$ ... it vanishes at some point $p$ when partial derivatives of $f$ at p are equal to zero (in some chart containing $p$ ).

Not quite sure what you mean, but if $w = df$ for $f\in C^{\infty}({\mathbb S}^2)$ , compactness of the sphere implies $f$ must attain two extrema (at least). So there are two points where $\omega=df=0$ .

Quote:

..is it correct to say

No.