Holonomy element from parallel transport around a loop

Let $\pi:P\to M$ be a principal $G$ -bundle with principal connection $\omega$ . Fix $x\in M$ , $p\in P_x$ , and a piecewise smooth loop $\gamma:[0,1]\to M$ with $\gamma(0)=\gamma(1)=x$ .

Using parallel transport, $p$ is carried to another point in the same fiber:

\mathrm{PT}^\omega_\gamma(p)\in P_x.

Because $P_x$ is a right $G$ -torsor, there is a unique element $h_\gamma(p)\in G$ such that

\mathrm{PT}^\omega_\gamma(p)=p\cdot h_\gamma(p).

This element is the holonomy element of $\omega$ along $\gamma$ based at $p$ .

Changing the point in the fiber conjugates the element: if $p'=p\cdot g$ , then

h_\gamma(p') = g^{-1}\,h_\gamma(p)\,g.

As $\gamma$ varies over based loops, the elements $h_\gamma(p)$ form the holonomy group at $p$ . Changing $p$ conjugates this subgroup, so its conjugacy class depends only on $x$ .

Examples

If $\omega$ is flat and $M$ is simply connected, then $h_\gamma(p)=e$ for all loops.
On the Levi-Civita connection of the round 2-sphere, transporting a tangent frame around a latitude circle yields a nontrivial rotation, giving a nontrivial holonomy element.
For an abelian structure group $G$ , the conjugation ambiguity disappears, so $h_\gamma(p)$ is independent of $p\in P_x$ .