Holonomy

Remarks

$\mathrm{Hol}_p(\nabla)$ is a group, because $P_{\gamma \cdot \delta} = P_\delta \circ P_\gamma$ and $P_{\gamma^{-1}} = P_\gamma^{-1}$. It is a Lie subgroup of $\mathrm{GL}(E_p)$, and $\mathrm{Hol}^0_p(\nabla)$ is its identity component; the quotient is a quotient of $\pi_1(M, p)$. Changing the base point along a curve $\sigma$ from $p$ to $q$ conjugates the group: $\mathrm{Hol}_q = P_\sigma \mathrm{Hol}_p P_\sigma^{-1}$. Example: the flat connection on $T\R^n$ has $P_\gamma = \id$ for every loop, so $\mathrm{Hol}_p$ is trivial. For the Levi-Civita connection of the round sphere $S^2$, $\mathrm{Hol}_p = \mathrm{SO}(2)$: the transport around a loop is the rotation by the enclosed area. For the Levi-Civita connection of any Riemannian $n$-manifold, $\mathrm{Hol}_p \subset \mathrm{O}(T_pM)$, and $\mathrm{Hol}_p \subset \mathrm{SO}(T_pM)$ if $M$ is orientable. Non-example: the flat connection on the Möbius line bundle over the circle has trivial restricted holonomy, but $\mathrm{Hol}_p = \set{\pm 1}$ is not trivial; so flatness alone does not force trivial holonomy.