# Restricted holonomy group

From Diffgeom

(Redirected from Restricted holonomy group of Riemannian metric)

## Definition

Let be a differential manifold, a vector bundle over , and a connection on . The **restricted holonomy group** at is defined as the subgroup of 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 normal 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).