  We study $n$ dimensional Riemanniann manifolds with harmonic forms of
constant length and first Betti number equal to $n-1$ showing that they are
2-steps nilmanifolds with some special metrics. We also characterise, in terms
of properties on the product of harmonic forms, the left invariant metrics
among them. This allows us to clarify the case of equality in the stable
isosytolic inequalities in that setting. We also discuss other values of the
Betti number.
