Holonomy of a loop

Definition

Let $M$ be a differential manifold, $E$ a vector bundle over $M$ and $\mathrm{\nabla }$ a connection on $E$. Let $m\in M$ and $\gamma :[0,1]\to M$ be a loop at $m$ (viz $\gamma (0)=\gamma (1)=m$. The holonomy of $\gamma$ is defined as the following linear transformation at $E_p$: it sends $v\in E_p$ to the element ${\varphi }_1(v)$ where ${\varphi }_t(v)$ is the transport of $v$ along $\gamma$, with respect to the connection $\mathrm{\nabla }$.

Related notions

• Holonomy group
• Restricted holonomy group