Chern–Weil theorem

Let $\pi:P\to M$ be a principal G-bundle with structure group $G$ and Lie algebra $\mathfrak g$ . Let $\omega$ be a principal connection with curvature $\Omega\in\Omega^2(P;\mathfrak g)$ .

Let $q$ be an $\operatorname{Ad}$ -invariant homogeneous polynomial of degree $k$ on $\mathfrak g$ , represented by a symmetric $k$ -linear map

q:\underbrace{\mathfrak g\times\cdots\times\mathfrak g}_{k\ \text{times}}\to \mathbb R \quad\text{satisfying}\quad q(\operatorname{Ad}(g)X_1,\dots,\operatorname{Ad}(g)X_k)=q(X_1,\dots,X_k).

Define the $2k$ -form on the total space $P$ by inserting $\Omega$ into $q$ and wedging:

q(\Omega)\in\Omega^{2k}(P),\qquad q(\Omega):=q(\Omega,\dots,\Omega),

with the usual graded antisymmetrization convention.

Theorem (Chern–Weil).

The form $q(\Omega)$ is closed, i.e. $d\,q(\Omega)=0$ .
The form $q(\Omega)$ is basic (see Chern–Weil forms are basic), hence by the basic forms theorem there is a unique closed form $\operatorname{cw}_q(\omega)\in\Omega^{2k}(M)$ with $\pi^*\operatorname{cw}_q(\omega)=q(\Omega).$
The de Rham cohomology class $[\operatorname{cw}_q(\omega)]\in H^{2k}_{\mathrm{dR}}(M)$ is independent of the choice of connection $\omega$ ; equivalently, changing $\omega$ changes $\operatorname{cw}_q(\omega)$ by an exact form (see the transgression theorem).

Remarks

A standard route to (1) is to combine the Bianchi identity with $\operatorname{Ad}$ -invariance of $q$ .

Examples

First Chern form for $U(1)$ . For $G=U(1)$ and $q(X)=\frac{i}{2\pi}X$ (viewing $\mathfrak u(1)\cong i\mathbb R$ ), $\operatorname{cw}_q(\omega)=\frac{i}{2\pi}F$ is the usual curvature representative of the first Chern class.
Second Chern character piece. For a matrix group such as $G=SU(n)$ and $q(X)=\operatorname{tr}(X^2)$ , the descended form $\operatorname{tr}(F\wedge F)$ is closed and defines a characteristic class independent of the connection.
Pontryagin-type forms. For $G=SO(n)$ , $\operatorname{Ad}$ -invariant polynomials built from traces of even powers, such as $\operatorname{tr}(X^{2j})$ in the defining representation, produce the usual Pontryagin form representatives on the base.

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