Holonomy group

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.

Related notions

• Restricted holonomy group
• Holonomy group of Riemannian metric