Geometrical formality of solvmanifolds and solvable Lie type geometries

Abstract

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 $\Gamma$, the solvmanifold $G/\Gamma$ 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.

Very important: The author is only one. to appear in RIMS Ko?kyu?roku Bessatsu: Geometry of Transformation Groups and Combinatorics