Parallel Transport - Path independence
Solution 1:
Here's a good place to use Gauss-Bonnet. If you parallel transport a vector around a small smooth closed curve $C$, the angle through which it turns is $2\pi - \int_C \kappa_g(s)\,ds$. It follows that $\iint_R K\,dA = 0$ for all small regions $R$. By continuity, $K=0$ everywhere.