Lemma: Chern–Weil forms are basic

Let $\pi:P\to M$ be a principal $G$ -bundle, let $\omega$ be a principal connection, and let $\Omega\in\Omega^2(P;\mathfrak g)$ be its curvature $2$ -form. If $Q$ is a real- or complex-valued Ad-invariant symmetric multilinear polynomial of degree $k$ on $\mathfrak g$ , then

Q(\Omega)=Q(\underbrace{\Omega,\dots,\Omega}_{k\text{ times}})\in\Omega^{2k}(P)

is a basic differential form, with the same real or complex coefficient field as $Q$ . Hence there is a unique $\alpha\in\Omega^{2k}(M)$ satisfying $\pi^*\alpha=Q(\Omega)$ .

Here substitution is defined directly: if $\Omega=\sum_a\Omega^a e_a$ in a basis of $\mathfrak g$ , then $Q(\Omega)=\sum_{a_1,\ldots,a_k}Q(e_{a_1},\ldots,e_{a_k})\,\Omega^{a_1}\wedge\cdots\wedge\Omega^{a_k}$ ; for $k=0$ it is the constant $Q$ . This expression is independent of the basis.

Horizontality and invariance

Horizontality: for every fundamental vertical vector field $X^\#$ on $P$ (see fundamental vector field convention), $\iota_{X^\#}\,Q(\Omega)=0.$ Equivalently, $Q(\Omega)$ vanishes whenever any argument is vertical.

$G$ -invariance: for every $g\in G$ , $R_g^*\,Q(\Omega)=Q(\Omega),$ so it is an invariant differential form.

Examples

Abelian case: $U(1)$ For $G=U(1)$ , the adjoint action is trivial, and $Q$ can be taken to be the identity on $\mathfrak{u}(1)\cong i\mathbb{R}$ . Then the Chern–Weil form is simply $Q(\Omega)=\Omega$ , and the lemma says $\Omega$ is basic, hence descends to a 2-form on $M$ . This is exactly what happens in the Dirac monopole example on the Hopf bundle.

Unitary bundles For a principal $U(n)$ -bundle, take $Q(X)=\mathrm{tr}(X)$ or $Q(X)=\mathrm{tr}(X^k)$ . The lemma guarantees that $\mathrm{tr}(\Omega)$ and $\mathrm{tr}(\Omega^k)$ are basic forms on $P$ , so they correspond to well-defined differential forms on $M$ whose normalized trace powers represent the components of the Chern character; the Chern classes are obtained from appropriate polynomial combinations of these normalized traces (see Chern class).

Orthogonal bundles and Pontryagin forms For $G=SO(n)$ , invariant polynomials such as $Q(X)=\mathrm{tr}(X^2)$ produce Pontryagin forms after the standard normalization (see Pontryagin class). The lemma ensures these forms are basic and hence live on the base manifold, not just on the total space of frames.