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Short question about Laplacians

  1. Sep 17, 2008 #1

    haushofer

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    Hi, I have a short question about Nakahara's treatment about Laplacian's: page 294, section 7.9.5, equation (7.188).

    He calculates the Laplacion [itex]\Delta = dd^{\dagger} + d^{\dagger}d [/itex] for a scalar function f. Every step is clear to me, except one; at the fourth line there is a factor of [itex]g^{-1} [/itex] popping up ( the determinant of the contravariant metric )

    What I get is ( ignoring the minus-sign in front )

    [tex]

    *d*df = * \frac{1}{(m-1)!} \partial_{\nu} [\sqrt{g}g^{\lambda\mu}\partial_{\mu}f]\epsilon_{\lambda\nu_{2}\cdots\nu_{m}} dx^{\nu}\wedge dx^{\nu_{2}}\wedge\ldots\wedge dx^{\nu_{m}}

    [/tex]

    just like Nakahara. Now I use

    [tex]

    dx^{\nu}\wedge dx^{\nu_{2}}\wedge \ldots \wedge dx^{\nu_{m}} = \epsilon^{\nu\nu_{2}\ldots\nu_{m}} dx^{1}\wedge \ldots \wedge dx^{m}

    [/tex]

    and the contraction

    [tex]
    \epsilon_{\lambda\nu_{2}\ldots\nu_{m}} \epsilon^{\nu\nu_{2}\ldots\nu_{m}} = (m-1)!\delta_{\lambda}^{\nu}
    [/tex]

    and simply fill this in. I get the same answer as is at line four of equation (7.188), except for that [tex]g^{-1}[/itex]. So I'm missing that determinant somewhere, but where?

    Many thanks in forward, my vision is a little blurred at the moment :)
     
    Last edited: Sep 17, 2008
  2. jcsd
  3. Sep 17, 2008 #2

    haushofer

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    I have the feeling that I should define

    [tex]

    dx^{\nu}\wedge dx^{\nu_{2}}\wedge \ldots \wedge dx^{\nu_{m}} = \epsilon_{\nu\nu_{2}\ldots\nu_{m}} dx^{1}\wedge \ldots \wedge dx^{m}

    [/tex]

    and that bringing those indices up on that epsilon tensor gives me that [itex]g^{-1}[/itex], but then the indices don't match.
     
  4. Oct 6, 2008 #3

    haushofer

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    I have another question, but it's rather short so I can kick this topic. If I have a principle fibre bundle P with base manifold M and fibre G and an associated vertical vector field X and projection pi: P to M, why exactly is the pullback of pi acting on X zero? Most of the Nakahara exercises are quite easy for me, but this one is troubling me for a day now. If I just apply the definition of a pullback to the vectical vector field, I don't see why it should be zero given the definition of this vertical vector field.
     
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