  We prove that manifolds admitting a Riemannian metric for which products of
harmonic forms are harmonic satisfy strong topological restrictions, some of
which are akin to properties of flat manifolds. Others are more subtle, and are
related to symplectic geometry and Seiberg-Witten theory.
  We also prove that a manifold admits a metric with harmonic forms whose
product is not harmonic if and only if it is not a rational homology sphere.
