Let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $\pi:P\to M$ be a principal $G$-bundle, and let $A$ be a principal connection on $P$ with curvature $F_A$. If $q\in I(G)$ is an $\operatorname{Ad}$-invariant polynomial on $\mathfrak g$, then the Chern-Weil characteristic form $q(F_A)\in \Omega^{\mathrm{even}}(M)$ is closed. Equivalently, for each homogeneous component $q_k\in I^k(G)$,

\begin{align*} d\bigl(q_k(F_A)\bigr)=0. \end{align*}