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Exterior derivatives on fibre bundles

  1. Jan 27, 2009 #1

    haushofer

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    Hi, I have a small question about exterior derivatives d on defined on principal bundles P.

    We have the Ehresmann connection on a principal bundle P, represented by a Lie-algebra valued one-form omega. We can use the section sigma to pull this one-form back to our basemanifold, where the interpretation of a gaugepotential comes in.

    Now one postulates this omega on P and checks if it fullfills the 2 axioms of this connection one-form. Now they (Nakahara chapter 10 ) use the following identification, which I don't see:

    [tex]
    d_{P} g_{i} (A^{*} ) = \frac{dg(ue^{tA})}{dt} \ |_{t=0}
    [/tex]

    Here [itex]g_{i}[/itex] is the canonical local trivialization,

    [tex]
    \phi_{i}^{-1}(p,g_{i}) = (p,g) \, \ \ \ u = \sigma_{i}(p) g_{i}
    [/tex]

    , A is an element of the Lie-algebra and [itex]A^{*}[/itex] is the fundamental vector field lying in the vertical subspace:

    [tex]
    A^{*}f(u) = \frac{d}{dt}f(ue^{tA}) \ |_{t=0}
    [/tex]

    Anyone an idea?
     
  2. jcsd
  3. Jan 27, 2009 #2

    haushofer

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    Okiedokie, solved the problem. :')

    Just an application of X(f) = df(X), so A(g) = dg(A).
     
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