Naturality of Chern–Weil classes under pullback

Let $\pi:P\to M$ be a principal G-bundle with a principal connection $\omega$ and curvature $\Omega$. 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

and hence the cohomology classes satisfy

Equivalently, on total spaces,

using the defining property of Chern–Weil forms .

Examples

1. Restriction to a submanifold. If $i:S\hookrightarrow M$ is an embedded submanifold, then $i^*P\to S$ carries the restricted characteristic classes: $\operatorname{cw}_P(i^*\omega)=i^*\operatorname{cw}_P(\omega)$.
2. 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.
3. 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 $k>0$.