We shall investigate Riemannian metrics on compact oriented manifolds for which all wedge products of harmonic forms are harmonic. In small dimensions, we aim to classify these metrics. In arbitrary dimensions, forms harmonic for such a metric have very special properties and define interesting geometric structures, like foliations and symplectic structures. We investigate homogeneous examples, and those close to homogeneous (e.g. of cohomogeneity one, or biquotients). Further we study constructions of such metrics via symplectic geometry. Within symplectic geometry, we study the analogous property for symplectically harmonic forms in the sense of Brylinski instead of the harmonic forms in the sense of Hodge theory.
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