covariant derivative

The covariant exterior derivative satisifes \(d_\nabla^2 s = F_\nabla \wedge s\) for \(s\in { { {\Omega}^{\scriptscriptstyle \bullet}} }_M(E)\) an \(E{\hbox{-}}\)valued form, and thus \(d^2=0\) when the curvature form vanishes. This yields a flat bundle.