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Thread: Smooth covector field on S^2

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    Smooth covector field on S^2

    Is there a smooth covector field on $\displaystyle {\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 $\displaystyle w = df $ for some smooth function on $\displaystyle {\mathbb S}^2 $... it vanishes at some point $\displaystyle p $ when partial derivatives of $\displaystyle f $ at p are equal to zero (in some chart containing $\displaystyle p $).

    ..is it correct to say that if we consider stereographic coordinates on $\displaystyle {\mathbb S}^2 $ then for any point $\displaystyle p $ where $\displaystyle w=df $ vanishes it will also vanish at point $\displaystyle -p $ ??
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    Super Member Rebesques's Avatar
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    Quote Originally Posted by Different View Post
    Is there a smooth covector field on $\displaystyle {\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 $\displaystyle w = df $ for some smooth function on $\displaystyle {\mathbb S}^2 $... it vanishes at some point $\displaystyle p $ when partial derivatives of $\displaystyle f $ at p are equal to zero (in some chart containing $\displaystyle p $).


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


    ..is it correct to say

    No.
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