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Non-vanishing vector field on S1xS2

  1. Feb 23, 2009 #1
    1. The problem statement, all variables and given/known data

    Hi everyone,

    This is a question on my tensor analysis/differential geometry homework due tomorrow, and I'm just not sure of the answer. The problem is to define a non-vanishing vector field V on S1 x S2.

    Part b of the question is to "sketch a nonvanishing vector field on the Klein bottle".

    2. Relevant equations

    A vector field V : M [tex]\rightarrow[/tex] TM is nonvanishing if V(p) is never equal to the zero tangent vector in TpM.

    So if I'm not mistaken, S1 x S2 = { x[tex]\in[/tex]R5 : x12 + x22 = 1, x32 + x42 + x52 =1 }

    3. The attempt at a solution

    So I figure, if I combine nonvanishing vector fields on both S1 andS2, I'll get one for the S1 x S2 manifold. I know that on S1, a nonvanishing vector field is just V(x1,x2) = (-x2, x1 ) because of the parametrization of a circle. However, a 2-sphere cannot be parametrized in terms of one coordinate, and even so, the hairy ball theorem states that it doesn't have a nonvanishing vector field. Will any vector field do, since the S1 vector field I'm using is already nonvanishing? The textbook (The Geometry of Physics by Frankel) doesn't anything about nonvanishing fields. Even if this is the case, I can't think of the parametrization that'll give me a good tangent field, since it has to give 0 when dotted with (x1,x2,x3)...

    For part b) of the question (the Kelin bottle), I'm pretty sure that a constant vector field in the direction parallel to the sides identified in the same direction on the rectangle will work, since it's a smooth manifold. However, if someone could confirm that this is right, I'd really appreciate it...

    Thanks for any help anyone offers!

    _______
    Alright, I had to hand it in, but if anyone knows the *right* answer, please do post it! I'll be going to office hours on thursday, probably, to find out anyways. My final answer was: since we can define a nonvanishing vector field on S1 V1 (x1, x2) -> (-x2,x1), and since there exists some vector field V2 tangent to S2, the combination of them V = (V1(x1, x2), V2(x3,x4,x5) ) will be nonvanishing since V1 is nonvanishing. Lame and vague/handwavy, but it's all I got.
     
    Last edited: Feb 24, 2009
  2. jcsd
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