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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
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