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Tangent Bundles, T(MxN) is Diffeomorphic to TM x TN

  1. Feb 26, 2013 #1
    1. The problem statement, all variables and given/known data

    If M and N are smooth manifolds, then T(MxN) is diffeomorphic to TM x TN

    2. Relevant equations



    3. The attempt at a solution

    So I'm here

    let ((p,q),v) [itex]\in[/itex] T(MxN)

    then p [itex]\in[/itex] M and q [itex]\in[/itex] N and v [itex]\in[/itex] T(p,q)(MxN).

    so T(p,q)(MxN) v = [itex]\sum_{i=1}^{m+n}[/itex] [itex]v_{i}\frac{\partial}{∂x_{i}}|_{(p,q)}[/itex]

    = [itex]\sum_{i=1}^{m}[/itex] [itex]v_{i}\frac{\partial}{∂x_{i}}|_{p}[/itex] + [itex]\sum_{i=}^{m+1}[/itex] [itex]v_{i}\frac{\partial}{∂x_{i}}|_{q}[/itex]

    not sure if this is it?
     
  2. jcsd
  3. Feb 26, 2013 #2

    micromass

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    You kind of assume that [itex]\frac{\partial}{\partial x^i}[/itex] are vector fields on both [itex]M\times N[/itex] and [itex]N[/itex]. I think you should be a little more careful than this...
     
  4. Feb 26, 2013 #3
    Hi Micro, I'm not sure I understand what you mean, am am assuming however that

    [itex]\sum_{i=1}^{m}[/itex] [itex]v_{i}\frac{\partial}{∂x_{i}}|_{p}[/itex] and [itex]\sum_{i=}^{m+1}[/itex] [itex]v_{i}\frac{\partial}{∂x_{i}}|_{q}[/itex]

    are bases for TM and TN respectively and I'm not sure if that's good enough.
     
  5. Feb 26, 2013 #4

    micromass

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    Well, you assume [itex]\frac{\partial}{\partial x_i}\vert_p[/itex] is both in [itex]T_pM[/itex] and [itex]T_pN[/itex]. You can't do that.
     
  6. Feb 26, 2013 #5
    v = [itex]\sum_{i=1}^{m}[/itex] [itex]v_{i}\frac{\partial}{∂x_{i}}|_{(p,q)}[/itex] + [itex]\sum_{i=}^{m+1}[/itex] [itex]v_{i}\frac{\partial}{∂x_{i}}|_{(p,q)}[/itex]

    so i need to find a map that takes v [itex]\rightarrow[/itex] (w,y)

    where w[itex]\in[/itex]TpM and y[itex]\in[/itex]TpN

    and (w,y) [itex]\in[/itex] TpM x TqN

    does this sound better? I was thinking about this problem a lot. I'm not sure if that's a diffeo right off the get.
     
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