# Exterior derivatives on fibre bundles

1. Jan 27, 2009

### 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:

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

Here $g_{i}$ is the canonical local trivialization,

$$\phi_{i}^{-1}(p,g_{i}) = (p,g) \, \ \ \ u = \sigma_{i}(p) g_{i}$$

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

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

Anyone an idea?

2. Jan 27, 2009

### haushofer

Okiedokie, solved the problem. :')

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