I am studying Lott's paper : "On the spectrum of a finite volume negatively-curved manifold" and the satement is following:
We have an compact infranilmanifold $N$ which is finitely covered by a nilmanifold $ \tilde{N} / \Gamma $ and endowed with a leftinvariant metric. On $N$ we have therefore a flat connection $\nabla$, such that all leftinvariant vectorfields on $N$ are parallel. Now it is said, that all harmonic forms on $N$ are parallel.
Why is this the case? It is clear, that every parallel form is harmonic, but why also the other direction?
Here is the link: Lott: On the spectrum of a fintie volume negatively-curved manifold It's in the proof of Theorem 1.
