# The holonomy group

We have seen that on a manifold with connection, the curvature measures the effect of a vector being parallel transported around an infinitesimal loop. If we consider the set of all closed loops at a basepoint $${p}$$ on a connected manifold with connection $${M^{n}}$$, the associated linear transformations due to parallel transport of a vector around each loop form a group called the holonomy group $${\mathrm{Hol}(M)}$$. The restricted holonomy group $${\mathrm{Hol}^{0}(M)}$$ only counts loops homotopic to zero.

From the definition of the parallel transporter we can see that $${\mathrm{Hol}(M)}$$ is in fact a group, and also a subgroup of $${GL(\mathbb{R}^{n})}$$, and therefore a Lie group. We can also see that for a connected manifold it is independent of the basepoint $${p}$$, since changing basepoints induces a similarity transformation (change of basis), altering the matrix representation of the group but acting as an isomorphism on the abstract group. If $${M}$$ is simply connected, then $${\mathrm{Hol}^{0}(M)=\mathrm{Hol}(M)}$$; if not, then $${\mathrm{Hol}^{0}(M)}$$ is the identity component of $${\mathrm{Hol}(M)}$$, which is a group representation of the fundamental group of $${M}$$ called the monodromy representation.

The above depicts how the holonomy group is comprised of elements $${\parallel_{L}\in GL(\mathbb{R}^{n})}$$ associated with parallel translation around loops; changing the basepoint of the loop from $${p}$$ to $${q}$$ induces a similarity translation $${\parallel_{C}^{-1}\parallel_{L}\parallel_{C}}$$, leaving the abstract group unchanged.

Since the curvature is the infinitesimal version of the holonomy construction, we might expect that it be related to the Lie algebra of $${\mathrm{Hol}(M)}$$, which is called the holonomy algebra. The Ambrose-Singer theorem confirms this; in the case of a simply connected manifold, it says that the Lie algebra of $${\mathrm{Hol}(M)}$$ is generated by all elements of the form $${\check{R}(v,w)\left|_{q\in M}\right.}$$.

Zero holonomy group then implies zero curvature; but the converse is only true for the restricted holonomy group, as can be seen by considering e.g. a flat sheet of paper rolled into a cone. However, zero curvature implies that the holonomy algebra vanishes, which means that the holonomy group is discrete.