Geometrical formality of solvmanifolds and solvable Lie type geometries. (English) Zbl 1295.53024

Kasuya, Hisashi

RIMS Kôkyûroku Bessatsu B39, 21-33 (2013).

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.

Reviewer: Andrew Bucki (Edmond)

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