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.
Primary Subjects: 53C25
Secondary Subjects: 53D35, 57R17, 57R57, 58A14
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