I'm writing a blog post on representation theory and I want to include some geometric example. Is this okay? "Let $M$ be a smooth connected manifold and let $\pi:E \to M$ be a vector bundle with a flat connection. Let $G=\pi_1(M)$ and $K=\mathbb{R}$. Let $x \in M$ be a base point.
If we take a smooth loop $ \gamma: S^1 \to \pi_1(M)$ based at $x$, parallel transport along that loop defines an automorphism of $V=T_xM$. The flatness condition gives us that this automorphism only depends on the homotopy class of $\gamma$ and by smooth approximation, every homotopy class of continuous loops may …