# Holonomy group

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## Definition

Let $M$ be a differential manifold, $E$ a vector bundle over $M$ and $\nabla$ a connection for $E$. For a point $m \in M$ the holonomy group at $m$ is the subgroup of $GL(E_p)$ comprising those linear transformations that arise as the holonomy of a loop at $m$.

If the differential manifold is path-connected, the holonomy groups at distinct points are conjugate as subgroups of $GL(E_p)$ so we can talk of the holonomy group.

## Facts

### Reduction of structure group to holonomy group

Any path-connected differential manifold can be treated as a principal bundle with structure group being the holonomy group. The description is as follows:

• Pick (arbitrarily) a basis at a particular point $p$
• Now, for each point $m$, the fibre at that point is the set of all bases at $m$ that can arise from the basis at $p$ by means of transport along a curve using the connection $\nabla$.
• In particular, any two bases at $m$ differ by the holonomy of a loop, which lies in $H$. Thus $H$ acts freely and transitively on the fibre at each point.

Moreover the holonomy group is the smallest group to which the structure group can be reduced (for the same reasons).