riemannian geometry - Is the wedge product of two harmonic forms harmonic?

Generically, the wedge product of two harmonic forms will not be harmonic. It is harder to find examples than counter-examples. For example, on compact Lie groups with a bi-invariant metric or, more generally, on riemannian symmetric spaces, harmonic forms are invariant and invariance is preserved by the wedge product. In general, though, this is not the case.

According to Kotschick (see, e.g., this paper) manifolds admitting a metric with this property are called geometrically formal and their topology is strongly constrained. He has examples, already in dimension 4, of manifolds which are not geometrically formal.