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

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.

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

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

Examples

1. 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 $\Omega=0$.

2. 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.

3. 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.