Lemma: Chern–Weil forms are basic

This lemma explains why the Chern–Weil construction produces differential forms on the base manifold from data on the total space.

Let $\pi:P\to M$ be a principal G-bundle , let $\omega$ be a principal connection on $P$, and let $\Omega\in\Omega^2(P;\mathfrak{g})$ be its curvature 2-form .

Let $P$ denote an $\mathrm{Ad}$-invariant symmetric multilinear polynomial of degree $k$ on $\mathfrak{g}$, so that the Chern–Weil form on the total space is

Statement

The form $P(\Omega)$ is basic on $P$ in the sense of basic differential forms on a principal bundle . Concretely:

1. Horizontality: for every fundamental vertical vector field $X^\#$ on $P$ (see fundamental vector field convention ),

   $$\iota_{X^\#}\,P(\Omega)=0.$$

   Equivalently, $P(\Omega)$ vanishes whenever any argument is vertical, so it is a horizontal form .

2. $G$-invariance: for every $g\in G$,

   $$R_g^*\,P(\Omega)=P(\Omega),$$

   so it is an invariant differential form .

Therefore there exists a unique form $\alpha\in\Omega^{2k}(M)$ such that $\pi^*\alpha=P(\Omega)$; this $\alpha$ is the Chern–Weil form on the base.

Proof idea

• The curvature $\Omega$ is horizontal: $\iota_{X^\#}\Omega=0$ for all fundamental vertical $X^\#$. Multilinearity of $P$ then implies $\iota_{X^\#}P(\Omega)=0$.
• The curvature transforms by the adjoint action: $R_g^*\Omega=\mathrm{Ad}_{g^{-1}}\Omega$ (using the adjoint action ). Since $P$ is $\mathrm{Ad}$-invariant, applying $P$ yields $R_g^*P(\Omega)=P(\Omega)$.

Examples

1. Abelian case: $U(1)$ For $G=U(1)$, the adjoint action is trivial, and $P$ can be taken to be the identity on $\mathfrak{u}(1)\cong i\mathbb{R}$. Then the Chern–Weil form is simply $P(\Omega)=\Omega$, 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.

2. Unitary bundles For a principal $U(n)$-bundle, take $P(X)=\mathrm{tr}(X)$ or $P(X)=\mathrm{tr}(X^k)$. 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 $M$ representing Chern classes (see Chern class ).

3. Orthogonal bundles and Pontryagin forms For $G=SO(n)$, invariant polynomials such as $P(X)=\mathrm{tr}(X^2)$ produce Pontryagin forms (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.