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 with right action $R_g:P\to P$, and let $A\in \Omega^1(P;\mathfrak g)$ be a principal connection with curvature form $F_A\in \Omega^2(P;\mathfrak g)$. Let $k\ge 0$, and let $P_0\in I^k(G)$, meaning that $P_0:\mathfrak g^k\to \mathbb R$ is a symmetric $k$-[linear map](/page/Linear%20Map) satisfying

\begin{align*} P_0(\operatorname{Ad}_g X_1,\dots,\operatorname{Ad}_g X_k)=P_0(X_1,\dots,X_k) \end{align*}

for all $g\in G$ and all $X_1,\dots,X_k\in \mathfrak g$.

Then the Chern-Weil form $P_0(F_A)\in \Omega^{2k}(P)$ is basic. Consequently, there exists a unique differential form $P_0(F_A)_M\in \Omega^{2k}(M)$ such that

\begin{align*} \pi^*P_0(F_A)_M=P_0(F_A). \end{align*}