TFAE: Flat principal bundles (principal G-bundle with connection)

Let $M$ be a connected smooth manifold and let $\pi:P\to M$ be a principal $G$ -bundle for a Lie group $G$ . Fix a principal connection $\omega$ on $P$ , with curvature form $\Omega$ . Then the following are equivalent:

Zero curvature. $\Omega=0$ ; equivalently, the curvature of $\omega$ vanishes identically.

Local horizontal sections. Every point of $M$ has a neighborhood $U$ admitting a smooth local section $s:U\to P$ such that $s^*\omega=0$ .

Homotopy-invariant parallel transport. Parallel transport along piecewise smooth curves depends only on the endpoint-fixed homotopy class of the curve. In particular, parallel transport around any contractible loop is the identity.

Trivial restricted holonomy. The restricted holonomy group, equivalently the identity component in the natural immersed Lie-group structure of the holonomy group, is trivial: $\mathrm{Hol}^0_p(\omega)=\{e\}$ for (equivalently, for every) $p\in P$ .

Classification by monodromy representation (when $M$ is connected). Choosing a basepoint $x\in M$ and $p\in P_x$ , there is a homomorphism (monodromy) $\rho:\pi_1(M,x)\to G$ such that $(P,\omega)$ is isomorphic (as a bundle with connection) to the quotient of the universal cover $\widetilde M\times G$ by the diagonal action of $\pi_1(M,x)$ given by deck transformations on $\widetilde M$ and left multiplication via $\rho$ on $G$ . The left multiplication commutes with the residual right principal $G$ -action.

Examples

Trivial bundle with the product (zero) connection. For $P=M\times G$ and the connection with horizontal distribution $TM\oplus\{0\}$ , one has $\Omega=0$ , parallel transport is constant in the $G$ -factor, and holonomy is trivial.

Flat bundles over the circle classified by a single element of $G$ . Over $M=S^1$ , any choice of $h\in G$ defines a flat bundle as the quotient $(\mathbb{R}\times G)/\mathbb{Z}$ where $1\in\mathbb{Z}$ acts by $(t,g)\mapsto(t+1,hg)$ . The curvature vanishes, and, with the base fiber represented by $[0,g]$ , parallel transport around the positively oriented generator sends $[0,g]$ to $[0,h^{-1}g]$ ; at $[0,e]$ its holonomy element is $h^{-1}$ . (Calling the quotient parameter $h^{-1}$ instead gives holonomy $h$ .)

Representations of the torus or surface group. A homomorphism $\rho:\pi_1(T^2)\to G$ (or $\pi_1(\Sigma_g)\to G$ for a surface) produces a flat principal bundle via the quotient $\widetilde M\times_\rho G$ . Distinct conjugacy classes of $\rho$ correspond to distinct flat bundles with connection up to isomorphism.