TITLE: Does curvature zero mean the bundle is trivial?
QUESTION [20 upvotes]: Let $P\to M$ be some Bundle over $M$. I know that, if $P$ is a trivial bundle it must have curvature zero. 
Say I have the converse, my curvature is zero. Does this imply that the bundle ist trivial?
If not, what can actually be said about the bundle, chern classes or it's connection if the curvature is zero? (I'm specifically interested in non-abelian principal G-bundles)

REPLY [11 votes]: Kobayashi-Nomizu discuss flat connections in Section II.9 of the first volume of their book and in particular derive the following as a corollary of earlier results.
Let $P \to M$ be a principal bundle equipped with a flat connection. If $M$ is simply connected, then $P$ is trivial.