Restricted holonomy group

Let $\pi:P\to M$ be a principal $G$-bundle with connection, and let $\mathrm{Hol}_p$ be the holonomy group at $p\in P$.

The restricted holonomy group at $p$, denoted $\mathrm{Hol}_p^0$, is the subgroup of $\mathrm{Hol}_p$ generated by parallel transport around loops $\gamma$ based at $x=\pi(p)$ that are contractible in $M$.

If $M$ is a connected smooth manifold , then $\mathrm{Hol}_p^0$ is a connected Lie subgroup of $G$, and it coincides with the identity component of $\mathrm{Hol}_p$. As with full holonomy, changing $p$ within the fiber conjugates $\mathrm{Hol}_p^0$.

Examples

1. Simply connected base. If $M$ is simply connected, every loop is contractible, hence $\mathrm{Hol}_p^0=\mathrm{Hol}_p$.

2. Flat connections. For a flat principal connection , holonomy depends only on the homotopy class of the loop. Contractible loops represent the identity element of $\pi_1(M)$, so $\mathrm{Hol}_p^0=\{e\}$, while $\mathrm{Hol}_p$ may still be nontrivial if $\pi_1(M)$ is nontrivial.

3. Discrete structure group. If $G$ is discrete (e.g. $G=O(1)=\{\pm1\}$), then the identity component of any subgroup is trivial. Thus $\mathrm{Hol}_p^0=\{e\}$ always, even when $\mathrm{Hol}_p$ is nontrivial.