Curvature 2-form of a principal connection

Let $\pi:P\to M$ be a principal $G$ -bundle with connection 1-form $\omega\in \Omega^1(P;\mathfrak{g})$ (see connection 1-form). The curvature 2-form is the $\mathfrak{g}$ -valued 2-form

\Omega \;\coloneqq\; d\omega \;+\; \tfrac12[\omega\wedge \omega] \;\in\; \Omega^2(P;\mathfrak{g}),

where $d$ is the exterior derivative and $[\omega\wedge\omega]$ is formed using the Lie bracket on $\mathfrak{g}$ together with the wedge product of 1-forms. The convention is $[\omega\wedge\omega](u,v)=2[\omega(u),\omega(v)]$ .

Properties

Key properties:

$\Omega$ is horizontal: $\Omega(X,\cdot)=0$ whenever $X$ is vertical.
$\Omega$ is equivariant: $R_g^*\Omega=\mathrm{Ad}(g^{-1})\Omega$ .

Consequently, $\Omega$ represents the curvature of the underlying principal connection and can be viewed as an $\mathrm{ad}(P)$ -valued 2-form on $M$ after descending from $P$ .

One conceptual characterization: if $X,Y$ are vector fields on $M$ and $X^H,Y^H$ are their horizontal lifts to $P$ , then

\Omega(X^H,Y^H) = -\,\omega([X^H,Y^H]),

so $\Omega$ measures the failure of horizontal lifts to close under brackets.

Examples

Maurer–Cartan form is flat. On $P=G\to\{\ast\}$ with $\omega=\theta=g^{-1}dg$ , the Maurer–Cartan equation says $d\theta+\tfrac12[\theta\wedge\theta]=0$ . Hence $\Omega=0$ .

Curvature in a trivialization. On $U\times G\to U$ with $\omega=\mathrm{Ad}(g^{-1})A+g^{-1}dg$ , one computes $\Omega = \mathrm{Ad}(g^{-1})\,\pi_U^*\!\left(dA+\tfrac12[A\wedge A]\right),$ so the entire curvature is pulled back from the base.

Abelian structure group. If $G$ is abelian, then $[\omega\wedge\omega]=0$ and $\Omega=d\omega$ . In particular for principal $U(1)$ -bundles the curvature is just the exterior derivative of the connection form.