Covariant derivative along a horizontal lift in an associated vector bundle

n.evans

I am trying to familiarize myself with the use of fibre bundles and associated bundles but am having some problems actually making calculations. I would like to show that the covariant derivative along the horizontal lift of a curve in the base space vanishes (which should be just a matter of employing definitions correctly I think):

Consider a principal fibre bundle [itex](E, \pi, M)[/itex] with a structure group [itex]G[/itex] associated to a vector bundle [itex](E_F, \pi_F, M)[/itex], where [itex]F[/itex] is a vector space. Let [itex]\alpha(t): [a, b] \rightarrow M[/itex] and [itex]\alpha^{\uparrow}_F(t): M \rightarrow E_F[/itex] be the horizontal lift of [itex]\alpha[/itex] in the associated bundle. Let [itex]\Psi(x): M \rightarrow E_F[/itex], with [itex]x \in M[/itex] be a section of the associated bundle, such that [itex]\Psi(\alpha(t)) = \alpha^{\uparrow}_F(t)[/itex]

Show that the covariant derivative [itex]\nabla_{\alpha}\Psi[/itex] evaluated along [itex]\alpha(t)[/itex] vanishes.

I know that the covariant derivative can be written as

[itex]\nabla_{\mu}\Psi(x) = \partial_{\mu}\Psi(x) + A_{\mu}(x)\Psi(x)[/itex]

but I cannot work out how to use the relation between [itex]\Psi[/itex] and [itex]\alpha^{\uparrow}_F(t)[/itex] to show that it vanishes (if indeed it should).