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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.

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  • $\begingroup$ It is good form to say which paper you are studying and, if possible, to include a link. This usually attracts more responses. $\endgroup$ Apr 27, 2014 at 0:18

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