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)$.

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