The shear construction

TY - JOUR

T1 - The shear construction

AU - Freibert, Marco

AU - Swann, Andrew

PY - 2019/2

Y1 - 2019/2

N2 - The twist construction is a method to build new interesting examples of geometric structures with torus symmetry from well-known ones. In fact it can be used to construct arbitrary nilmanifolds from tori. In our previous paper, we presented a generalization of the twist, a shear construction of rank one, which allowed us to build certain solvable Lie algebras from R n via several shears. Here, we define the higher rank version of this shear construction using vector bundles with flat connections instead of group actions. We show that this produces any solvable Lie algebra from R n by a succession of shears. We 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 (h 3 ⊕ R 3 ) ⋊ R.

AB - The twist construction is a method to build new interesting examples of geometric structures with torus symmetry from well-known ones. In fact it can be used to construct arbitrary nilmanifolds from tori. In our previous paper, we presented a generalization of the twist, a shear construction of rank one, which allowed us to build certain solvable Lie algebras from R n via several shears. Here, we define the higher rank version of this shear construction using vector bundles with flat connections instead of group actions. We show that this produces any solvable Lie algebra from R n by a succession of shears. We 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 (h 3 ⊕ R 3 ) ⋊ R.

KW - Calibrated and cocalibrated G2-structures

KW - Generalization of twist construction

KW - Solvable Lie groups

UR - http://www.scopus.com/inward/record.url?scp=85041860983&partnerID=8YFLogxK

UR - https://arxiv.org/abs/1702.05318

U2 - 10.1007/s10711-018-0330-9

DO - 10.1007/s10711-018-0330-9

M3 - Journal article

AN - SCOPUS:85041860983

SN - 0046-5755

VL - 198

SP - 71

EP - 101

JO - Geometriae Dedicata

JF - Geometriae Dedicata

IS - 1

ER -