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 $P$ be an Ad-invariant homogeneous polynomial of degree $k$ on $\mathfrak g$, equivalently a symmetric $k$-linear map

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

with the usual graded antisymmetrization convention.

Theorem (Chern–Weil).

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

A standard route to (1) is to combine the Bianchi identity with Ad-invariance of $P$.

Examples

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