Geometrical formality of solvmanifolds and solvable Lie type geometries

We show that for a Lie group $G=\R^{n}\ltimes_ϕ \R^{m}$ with a semisimple action $ϕ$ which has a cocompact discrete subgroup $Γ$, the solvmanifold $G/Γ$ admits a canonical invariant formal (i.e. all products of harmonic forms are again harmonic) metric. We show that a compact oriented aspherical manifold of dimension less than or equal to 4 with the virtually solvable fundamental group admits a formal metric if and only if it is diffeomorphic to a torus or an infra-solvmanifold which is not a nilmanifold.

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