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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:
<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?
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?
