The shear construction

Marco Freibert*, Andrew Swann

*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review


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.

Original languageEnglish
JournalGeometriae Dedicata
Pages (from-to)71-101
Number of pages31
Publication statusPublished - Feb 2019


  • Calibrated and cocalibrated G2-structures
  • Generalization of twist construction
  • Solvable Lie groups


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