Isomorphic principal bundles have the same Chern–Weil classes

Let $P\to M$ and $P'\to M$ be principal $G$ -bundles, and let $\Phi:P\to P'$ be a principal bundle isomorphism over $M$ (a $G$ -equivariant diffeomorphism with $\pi'\circ\Phi=\pi$ ).

For every invariant polynomial $p$ , the corresponding Chern–Weil characteristic class satisfies

\mathrm{cw}_p(P)=\mathrm{cw}_p(P') \in H^{*}_{\mathrm{dR}}(M).

Equivalently: choosing any connection $\omega$ on $P$ , the pullback connection $(\Phi^{-1})^*\omega$ on $P'$ has the property that the Chern–Weil forms produced from $\omega$ and $(\Phi^{-1})^*\omega$ represent the same de Rham class, so the classes coincide.

Examples

Same bundle via different constructions. If a principal bundle is presented using two different atlases or two different cocycles related by a coboundary, the resulting bundles are isomorphic; their Chern–Weil classes coincide.
Vector bundle isomorphism. If two rank- $n$ real vector bundles are isomorphic, their frame bundles are isomorphic as principal $GL(n,\mathbb R)$ -bundles, so all Chern–Weil classes defined from the frame bundle agree (e.g. Pontryagin classes).
Pullback by a diffeomorphism. If $f:N\to M$ is a diffeomorphism and $P\to M$ is a principal bundle, then $f^*P\to N$ is isomorphic to $P$ transported along $f$ , and the Chern–Weil classes transform by pullback in cohomology.