# zbMATH — the first resource for mathematics

Geometrical formality of solvmanifolds and solvable Lie type geometries. (English) Zbl 1295.53024
A Riemannian metric $$g$$ on a compact oriented manifold $$M$$ is called formal if all products of harmonic forms are harmonic and $$M$$ is called geometrical formal.
In this paper, the author studies the geometric formality of solvmanifolds and solvable Lie type geometries. It is shown that if $$G=\mathbb R^n\ltimes_\varphi\mathbb R^m$$ is a Lie group with a semisimple action $$\varphi$$ which has a cocompact discrete subgroup $$\Gamma$$ and $$G / \Gamma$$ is the associated solvmanifold, then $$G/\Gamma$$ admits a canonical invariant formal metric. Also, the author proves that if $$M$$ is a compact oriented aspherical manifold of dimension less than or equal to $$4$$ with the virtually solvable fundamental group, then $$M$$ is geometrically formal if and only if $$M$$ is diffeomorphic to a torus or an infra-solvmanifold which is not a nilmanifold.

##### MSC:
 53C20 Global Riemannian geometry, including pinching 22E25 Nilpotent and solvable Lie groups 58A14 Hodge theory in global analysis 53C30 Differential geometry of homogeneous manifolds
##### Keywords:
solvmanifold; harmonic form; geometrical formal manifold
Full Text: