Geometrical formality of solvmanifolds and solvable Lie type geometries

We show that for a Lie group $G={\mathbb{R}}^n{⋉}_{\varphi }{\mathbb{R}}^m$ with a semisimple action $\varphi$ which has a cocompact discrete subgroup $\mathrm{\Gamma }$, the solvmanifold $G\mathrm{/}\mathrm{\Gamma }$ 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.