Restricted holonomy group

Definition

Let $M$ be a differential manifold, $E$ a vector bundle over $M$, and $\mathrm{\nabla }$ a connection on $E$. The restricted holonomy group at $m\in M$ is defined as the subgroup of $GL(E_m)$ comprising all those linear transformations that occur as the holonomy of a loop that is homotopic to the constant loop.

Facts

The restricted holonomy group is a subgroup of the holonomy group, which comprises all the linear transformations that occur as the holonomy of a loop (not necessarily homotopic to the constant loop).