We begin this lecture by wrapping up our introductory account of connections. We then move on to holonomy, which—roughly speaking—measures the global dependence of the parallel transport maps $\widehat{\mathbb{P}}_{ \gamma}$ on the given path $\gamma$.

This allows us to give a “trivialisation-free” definition of what it means for a connection to be trivial, and prove that only trivial vector bundle admit trivial connections.

Finally we sketch why the holonomy group $\operatorname{Hol}^{\nabla}(x)$ is a Lie subgroup of $\operatorname{GL}(E_x)$.