Flat principal connection

Let $\pi:P\to M$ be a principal $G$-bundle equipped with a principal connection and curvature $\Omega\in \Omega^2(P;\mathfrak{g})$.

The connection is called flat if its curvature 2-form vanishes:

Equivalently, in any local trivialization with local connection form $A$, the local curvature satisfies $F=dA+\tfrac12[A\wedge A]=0$.

A flat connection has the key consequence that parallel transport along a curve depends only on the homotopy class of the curve (with endpoints fixed). In particular, parallel transport around loops yields the holonomy representation of $\pi_1(M)$ into $G$, and the associated holonomy group is determined by the image of that representation.

Examples

1. Product connection on a trivial bundle. On $P=M\times G$, take the connection defined by $\omega=g^{-1}dg$ (equivalently $A=0$ in the canonical trivialization). Then $\Omega=0$, so the connection is flat and parallel transport leaves the $G$-coordinate unchanged.

2. Flat bundle from a representation. Given a representation $\rho:\pi_1(M)\to G$, the quotient $P=(\widetilde M\times G)/\pi_1(M)$ (with $\gamma\in\pi_1$ acting by deck transformations on $\widetilde M$ and by right multiplication through $\rho(\gamma)^{-1}$ on $G$) carries a natural flat connection descended from the product connection on $\widetilde M\times G$.

3. Constant commuting gauge field on a torus. On $T^n$ with a trivial $G$-bundle, a connection form $A=\sum_i A_i\,d\theta^i$ with constant $A_i\in\mathfrak{g}$ is flat provided $[A_i,A_j]=0$ for all $i,j$ (so both $dA=0$ and $[A\wedge A]=0$).