Naturality of Chern–Weil classes under pullback

Let $\pi:P\to M$ be a principal G-bundle with a principal connection $\omega$ and curvature 2-form $\Omega$ . Let $f:N\to M$ be a smooth map and let $f^*P\to N$ be the pullback principal bundle, with pulled-back connection $f^*\omega$ and curvature $f^*\Omega$ .

Fix an Ad-invariant homogeneous polynomial $P$ on $\mathfrak g$ of degree $k$ .

Theorem (Naturality). The Chern–Weil forms satisfy

f^*\big(\operatorname{cw}_P(\omega)\big)=\operatorname{cw}_P(f^*\omega),

and hence the cohomology classes satisfy

f^*\big([\operatorname{cw}_P(\omega)]\big)=[\operatorname{cw}_P(f^*\omega)]\in H^{2k}_{\mathrm{dR}}(N).

Equivalently, on total spaces,

(f^*\pi)^*\operatorname{cw}_P(f^*\omega)=P(f^*\Omega)=f^*(P(\Omega))=f^*\big(\pi^*\operatorname{cw}_P(\omega)\big),

using the defining property of Chern–Weil forms.

Examples

Restriction to a submanifold. If $i:S\hookrightarrow M$ is an embedded submanifold, then $i^*P\to S$ carries the restricted characteristic classes: $\operatorname{cw}_P(i^*\omega)=i^*\operatorname{cw}_P(\omega)$ .
Diffeomorphism invariance. If $f$ is a diffeomorphism, then characteristic classes transform by pullback under $f$ and in particular are invariants of the bundle up to isomorphism over the diffeomorphic base.
Constant map. If $f:N\to M$ is constant, then $f^*P$ is a trivial bundle; the pulled-back characteristic classes are zero in positive degree, so $f^*([\operatorname{cw}_P(\omega)])=0$ for $k>0$ .