Chern–Weil classes are independent of the connection

Let $\pi:P\to M$ be a principal $G$ -bundle, and let $p$ be an $\mathrm{Ad}$ -invariant polynomial on the Lie algebra (for example, a symmetric $k$ -linear form on $\mathfrak g$ invariant under the adjoint action). Given a principal connection $\omega$ with curvature $\Omega$ , the Chern–Weil construction produces a differential form on $M$ by applying $p$ to $\Omega$ and using the fact that the resulting form is basic.

Corollary (independence of connection)

For each invariant polynomial $p$ of degree $k$ , there is a canonically defined de Rham cohomology class

\mathrm{cw}_p(P)\in H^{2k}_{\mathrm{dR}}(M)

with the following property:

For any principal connection $\omega$ on $P$ , the Chern–Weil form $\mathrm{CW}_p(\omega)\in \Omega^{2k}(M)$ satisfies $d\,\mathrm{CW}_p(\omega)=0$ (closedness, using the exterior derivative), and its class $[\mathrm{CW}_p(\omega)]$ depends only on $P$ , not on $\omega$ .

Equivalently, if $\omega_0,\omega_1$ are two connections on $P$ , then

\mathrm{CW}_p(\omega_1)-\mathrm{CW}_p(\omega_0)

is an exact differential form on $M$ . Thus Chern–Weil characteristic classes are invariants of the underlying principal bundle.

Examples

First Chern class of a line bundle. For a principal $U(1)$ -bundle (complex line bundle), choosing $p$ to be the identity on $\mathfrak u(1)$ gives a closed 2-form representing the first Chern class in real cohomology; changing the connection changes the representative by an exact form.
Pontryagin classes. For a principal $SO(n)$ -bundle, invariant polynomials built from traces of powers of curvature produce the Pontryagin classes. For $TM$ , this shows Pontryagin classes are independent of the chosen Riemannian metric and its Levi–Civita connection.
Second Chern class for $SU(2)$ . For a principal $SU(2)$ -bundle over a 4-manifold, the invariant polynomial $p(X,Y)=\mathrm{tr}(XY)$ yields a 4-form representing the second Chern class (instanton number), independent of the chosen connection.