For a Lie group $G$, let $\pi: E \rightarrow M$ be a principal $G$-bundle over a smooth manifold $M$. Let $\omega: TE \rightarrow Lie(G)$ be a connection 1-form on $E$. Let $f:N \rightarrow M$ be smooth map. We know that there is a natural principal $G$-bundle structure on the pullback bundle $f^{*}E \rightarrow N$, induced from the principal $G$– bundle $\pi:E \rightarrow M$.
My question is the following:
Is there a natural way to define a connection 1-form on the pullback G-bundle $f^{*}E$ for any smooth map $f:N \rightarrow M$, induced from $\omega$? If not, then under what condition on $f:N \rightarrow M$, we can naturally define a connection 1-form on $f^{*}E$, induced from $\omega$?
Edit:[This question is closed] I apologise deeply for asking the question in this forum without providing any context. This question came up to me in the context of constructing a stack of principal $G$-bundles with connections over smooth manifolds. For the construction of the associated pseudofunctor, I had to ensure a canonical connection form on the pullback principal bundle induced from a connection form on the original principal bundle.
I will try my best to not repeat these kind of mistakes while asking questions in the forum in future.
Best Answer
The pullback bundle $f^*E$ satisfies the following commutative diagram: $\require{AMScd}$ \begin{CD} f^*E @>{F}>> E\\ @V{\pi}VV @VV{\pi}V\\ N @>{f}>> M \end{CD} So we can pull the form $\omega\in \Omega^1(E,\mathrm{Lie}(G))$ back to a form on $F^*\omega\in \Omega^1(f^*E,\mathrm{Lie}(G))$ defined by $F^*\omega(v)=\omega(T_p F v)$ for $v\in T_pf^*E$. Since the $G$ action on $f^*E$ comes from the $G$ action on $E$, this form will define a connection form.