Horizontal distribution of a principal connection ($H=\ker \mathcal{A}$)

The horizontal distribution of a principal connection is the smooth complement $H$ to the tangent spaces of the group orbits. A tangent vector is horizontal exactly when the connection form $\mathcal{A}$ annihilates it.

A local section $s:U\to P$ is horizontal when $ds(TU)\subseteq H$, equivalently when $s^{\ast }\mathcal{A}=0$. A flat principal connection has horizontal sections locally, while its holonomy can obstruct a global horizontal section.

The holonomy of a connection along a closed curve is the group element relating the endpoints of its horizontal lift. A flat connection can have nontrivial holonomy around a noncontractible loop.