Holonomy group

Let $\pi:P\to M$ be a principal $G$-bundle equipped with a principal connection . Fix a point $p\in P$ and set $x=\pi(p)\in M$.

For a piecewise smooth loop $\gamma:[0,1]\to M$ with $\gamma(0)=\gamma(1)=x$, let $\widetilde\gamma:[0,1]\to P$ be the horizontal lift starting at $p$ (i.e. $\widetilde\gamma(0)=p$ and $\dot{\widetilde\gamma}(t)\in H_{\widetilde\gamma(t)}$ for all $t$). Since $\widetilde\gamma(1)$ lies in the same fiber as $p$, there is a unique $g_\gamma\in G$ such that

The holonomy group at $p$ is

Equivalently, $\mathrm{Hol}_p$ is the subgroup generated by all parallel transports along loops based at $x$.

If $p' = p\cdot h$ is another point in the same fiber, then $\mathrm{Hol}_{p'} = h^{-1}\mathrm{Hol}_p\,h$; thus the holonomy group is well-defined up to conjugacy inside $G$.

Examples

1. Trivial flat connection. On $M\times G$ with the flat product connection, horizontal lifts keep the $G$-coordinate constant, so $\mathrm{Hol}_p=\{e\}$.

2. Flat connection over $S^1$. For a flat connection on a bundle over $S^1$, parallel transport around the fundamental loop yields an element $h\in G$; the holonomy group is the subgroup generated by $h$ (and is therefore cyclic if $G$ is discrete, or may be a torus-type subgroup in a compact Lie group).

3. Hopf fibration. For the standard connection on the Hopf bundle $S^3\to S^2$ with structure group $U(1)$, holonomy around loops in $S^2$ produces arbitrary phases, so $\mathrm{Hol}_p = U(1)$.