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Suppose we have a principal $G$-bundle $\pi:P\rightarrow M$ with a connection $\omega:TP\rightarrow \mathfrak{g}$, where $\mathfrak{g}$ is the lie algebra of $G$. Now suppose I have an open set $U$ in $M$ such that there is a local trivialisation $\phi:\pi^{-1}(U)\rightarrow U\times G$, with $ \phi(p)=(\pi(p), g)$.

Can anyone explain why the locally we have

$(\phi^{-1})^*\omega_{(\pi(p), g)}=Ad_{g^{-1}}(a_U) +g^{-1}dg $,

where $a_U$ is a lie algebra valued one form on $U$ and $g^{-1}dg$ is Maurer-Cartan form.

I would really appreciate help on this,

Thanks.

Also if anything is unclear please tell me.

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$\def\Ad{\mathrm{Ad}}$First note that $g^{-1} dg$ is a principal connection on $U \times G.$ Thus $$\eta = (\phi^{-1})^* \omega - g^{-1}dg \in \Omega^1(U\times G;\mathfrak g)$$ is a difference of principal connections, so it is equivariant and horizontal, meaning $\Ad_g (R_g^* \eta) = \eta$ and $\eta|_{TG} = 0.$ Using these two properties, we can determine $\eta$ from its values on $TU \times \{e \}:$ $$\eta_{x,g} (v, \xi) = \eta_{x,g}(v,0)=\Ad_{g^{-1}}(R_{g^{-1}}^*(\eta_{x,e})(v,0)).$$ But $R_{g^{-1}}^* (\eta_{x,e})(v,0) = \eta_{x,e}(DR_{g^{-1}} v,0)=\eta_{x,e}(v,0)$ since the group multiplication acts only on the $G$ factor; so we have $\eta = \Ad_{g^{-1}}(a_U)$ with $a_U = \eta|_{TU \times \{e \}}.$

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  • $\begingroup$ Thank you, really appreciate your help. $\endgroup$ – Speirsy Nov 10 '17 at 1:08

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