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Tensor product commutes with pullback?

  1. Feb 24, 2008 #1

    CompuChip

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    Hello,

    I have an exercise where we have to pullback a metric [itex]g^{ij} \, \mathrm dx_i \, \mathrm dx_j[/itex] under a function [itex]f: M \rightarrow N[/itex] (actually in this case [itex]M = \mathbf{R}^2, N = \mathbf{R}^3[/itex], but that's not really relevant).

    I managed to do it, provided that the pullback commutes with the tensor product. That is, using that actually [itex]\mathrm dx_i \, \mathrm dx_j = \mathrm dx_i \otimes \mathrm dx_j[/itex], I assumed that
    [tex]f^*(\mathrm dx_i \, \mathrm dx_j) = (f^*(\mathrm dx_i)) \otimes (f^*(\mathrm dx_j))[/tex]
    so then I could use that
    [tex]f^*(\mathrm dx_i) = \mathrm d(f^* x_i) = \mathrm df_i = \frac{\partial f_i}{\partial x_k} \mathrm dx_k [/tex]
    and finish the exercise.

    Why is this true?

    Thanks a lot.
     
  2. jcsd
  3. Feb 24, 2008 #2

    Fredrik

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    [tex]f^*(dx_i\otimes dx_j)(u,v)=dx_i\otimes dx_j (f_*u,f_*v)=dx_i(f_*u) dx_j(f_*v) [/tex]

    [tex]=f^*(dx_i)(u)f^*(dx_j)(v)=f^*(dx_i)\otimes f^*(dx_j)(u,v)[/tex]
     
  4. Feb 25, 2008 #3

    CompuChip

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    Thank you very much!

    I missed a lot on this topic though, so I'm not really familiar with pulls/pushes. Could you clarify the first (= third) step a bit?
    Basically, it doesn't matter whether we push u to TN and apply dx, or if we pull dx to T*M and apply it to u -- is this just the "d commutes with push-forwards/pullbacks" statement?

    [edit]I found some info here, it seems to be pretty elementary.

    Anyway, thanks again for the help. [/edit]
     
    Last edited: Feb 25, 2008
  5. Feb 25, 2008 #4

    Fredrik

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    Every step in my calculation is a simple application of a definition, of either the tensor product or the pullback. If you meant the third equality sign, I'm just using the definition of a pullback there. I suppose I could have dropped some of the parentheses though and wrote the second line as

    [tex]=f^*dx_i(u)f^*dx_j(v)=f^*dx_i\otimes f^*dx_j(u,v)[/tex]
     
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