Convention: local curvature is F = dA + A wedge A

A principal connection can be described locally by a Lie algebra–valued 1-form, and its curvature is then expressed by a standard structure equation. Different sign conventions appear in the literature; this knowl fixes the convention used here.

Let $\pi:P\to M$ be a principal G-bundle and let $\omega\in\Omega^1(P;\mathfrak{g})$ be a connection 1-form . Let $\Omega\in\Omega^2(P;\mathfrak{g})$ be its curvature (see curvature 2-form of a principal connection ).

Fix a local section $s:U\to P$ over an open set $U\subseteq M$. Define:

• the local connection 1-form on $U$ by $A := s^*\omega \in \Omega^1(U;\mathfrak{g}),$ as in local connection 1-form ;
• the local curvature 2-form on $U$ by $F := s^*\Omega \in \Omega^2(U;\mathfrak{g}),$ as in local curvature 2-form .

Convention

We use the local curvature formula

This is the convention recorded in the local curvature formula F = dA + A wedge A .

Here $d$ is the exterior derivative , and $A\wedge A$ is the $\mathfrak{g}$-valued 2-form defined by

using the Lie bracket on $\mathfrak{g}$. (This is a standard way to combine the wedge product on forms with the bracket on the Lie algebra.)

With this convention, the global structure equation on $P$ can be written in the parallel form

where $\omega\wedge\omega$ uses the same bracketed wedge construction.

Examples

1. Abelian structure group If $G$ is abelian (for instance $U(1)$), then $[\,\cdot,\cdot\,]=0$ on $\mathfrak{g}$, so $A\wedge A=0$ and the formula reduces to

   $$F=dA.$$

2. Pure gauge gives zero curvature On a trivial bundle over $U$, a pure gauge local connection can be written as

   $$A = g^{-1}dg$$

   for a smooth map $g:U\to G$ (compare pure gauge connection ). With the convention above, one obtains

   $$F = dA + A\wedge A = 0,$$

   which is the local manifestation of flatness.

3. Non-abelian contribution from A wedge A For a non-abelian $G$ (for example $SU(2)$), even if the components of $A$ have constant coefficients in a coordinate chart, the term $A\wedge A$ can be nonzero because it depends on the Lie bracket. This is the origin of genuinely non-linear curvature effects in gauge theory.