The shear construction (Q1737573)

The twist construction is a method to build new examples of geometric structures with torus symmetry from well-known ones. It can be used to construct arbitrary nilmanifolds from tori. In a previous work by the authors [J. Geom. Phys. 106, 268--274 (2016; Zbl 1341.53054)], they presented a generalization of the twist, a shear construction of rank one, which allows to build certain solvable Lie algebras from \(\mathbb{R}^n\) via several shears. In the present paper, they define higher-rank version of this shear construction using vector bundles with flat connections instead of group actions. They show that this produces any solvable Lie algebra from \(\mathbb{R}^n\) by a succession of shears. They give examples of the shear and discuss in detail how one can obtain certain geometric structures (calibrated \(G_2\), co-calibrated \(G_2\) and almost semi-Kähler) on three-step solvable Lie algebras by shearing almost abelian Lie algebras. This discussion yields a classification of calibrated \(G_2\)-structures on Lie algebras of the form \((\mathfrak{h}_3\oplus \mathbb{R}^3)\rtimes\mathbb{R}\).