Let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $\pi:P\to M$ be a smooth principal $G$-bundle admitting a principal connection, and let $f:N\to M$ be a smooth map. Let $f^*P\to N$ denote the pullback principal $G$-bundle. Then $f^*P$ admits the pulled-back principal connection, and for every invariant polynomial $P_0\in I(G)$ one has

\begin{align*} \operatorname{cw}_{f^*P}(P_0)=f^*\operatorname{cw}_{P}(P_0) \end{align*}

in $H^{\mathrm{even}}_{\mathrm{dR}}(N)$.