Exterior derivatives on fibre bundles

[haushofer]

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:

<br /> d_{P} g_{i} (A^{*} ) = \frac{dg(ue^{tA})}{dt} \ |_{t=0}<br />

Here g_{i} is the canonical local trivialization,

<br /> \phi_{i}^{-1}(p,g_{i}) = (p,g) \, \ \ \ u = \sigma_{i}(p) g_{i}<br />

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

<br /> A^{*}f(u) = \frac{d}{dt}f(ue^{tA}) \ |_{t=0}<br />

Anyone an idea?

 

[haushofer]

Okiedokie, solved the problem. :')

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