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

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

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

Examples

1. 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.
2. 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.
3. 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.