Vector Bundle Over Contractible Manifold

Solution 1:

When your manifold is not contractible, the parallel transport depends on the path you choose. If you take a flat connection, it only depends on the homotopy class of the path you choose. The parallel transport is always smooth because you solve localy a second degree differential equation...

If you want to see it in the case of the torus: take a vector bundle $E$ and a flat connection (with non trivial holonomy) you see that if you take a small loop $ \gamma$ around a given point $x$, the parallel transport will give the identity of $E_{x}$ but if you follow the meridian or the longitude you will have a non trivial Automorphism of $E_{x}$.