Local curvature 2-form

Let $\pi:P\to M$ be a principal $G$-bundle with connection form $\omega$ and curvature $\Omega$ as in the curvature 2-form of a principal connection .

On an open set $U\subset M$ with local section $s:U\to P$, define the local connection 1-form $A=s^*\omega\in \Omega^1(U;\mathfrak{g})$. The local curvature 2-form on $U$ is

Then $F$ is given by the structure equation

where $d$ is the exterior derivative and the bracket uses the Lie bracket on $\mathfrak{g}$.

If $s' = s\cdot g$ is another local section related by a gauge function $g:U\to G$, then the associated local curvature forms satisfy the gauge transformation rule

compatible with the gauge transformation of the local connection form .

Examples

1. Abelian case. If $G$ is abelian (e.g. $U(1)$), then $[A\wedge A]=0$, hence $F=dA$, and the gauge rule reduces to $F'=F$.

2. Matrix-valued form convention. For $G\subset GL(n)$, one often writes $F=dA + A\wedge A$. This matches the formula above because for matrix-valued 1-forms, $[A\wedge A]=A\wedge A - (A\wedge A)^{\mathrm{op}} = 2A\wedge A$.

3. Trivial (flat) local connection. If $A=0$ on $U$, then $F=0$ on $U$, so the connection is locally flat there.