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Math Help - Smooth covector field on S^2

  1. #1
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    Smooth covector field on S^2

    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 ??
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  2. #2
    Super Member Rebesques's Avatar
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    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.


    ..is it correct to say

    No.
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