Let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $M$ be a smooth manifold, and let $\pi:P\to M$ be a smooth principal $G$-bundle admitting a principal connection. For each integer $k\ge 0$, let

\begin{align*} I^k(G):=(\operatorname{Sym}^k\mathfrak g^*)^G \end{align*}

denote the [vector space](/page/Vector%20Space) of $\operatorname{Ad}$-invariant symmetric $k$-linear forms on $\mathfrak g$, with $I^0(G)=\mathbb R$, and let

\begin{align*} I(G):=\bigoplus_{k\ge 0} I^k(G) \end{align*}

be the corresponding graded algebra of invariant polynomials.

For a principal connection $A\in\Omega^1(P;\mathfrak g)$ with curvature $F_A\in\Omega^2(P;\mathfrak g)$ and for $\Phi\in I^k(G)$, let $\Phi(F_A)_M\in\Omega^{2k}(M)$ denote the unique descended differential form whose pullback to $P$ is the Chern-Weil form $\Phi(F_A)\in\Omega^{2k}(P)$. Then $\Phi(F_A)_M$ is closed, and the de Rham cohomology class

\begin{align*} [\Phi(F_A)_M]\in H^{2k}_{\mathrm{dR}}(M) \end{align*}

is independent of the chosen principal connection $A$.

Consequently the assignment $\operatorname{cw}_P:I(G)\to H^{\mathrm{even}}_{\mathrm{dR}}(M)$ defined on homogeneous elements by

\begin{align*} \operatorname{cw}_P(\Phi):=[\Phi(F_A)_M] \end{align*}

is a well-defined graded algebra homomorphism, sending polynomial degree $k$ to de Rham degree $2k$.