Advertisement

The shear construction

  • Marco Freibert
  • Andrew Swann
Original Paper
  • 50 Downloads

Abstract

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 \({\mathbb R}^n\) 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 \({\mathbb R}^n\) 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 \((\mathfrak {h}_3\oplus {\mathbb R}^3)\rtimes {\mathbb R}\).

Keywords

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

Mathematics Subject Classification

Primary 53C15 Secondary 53C12 22E25 32C37 

Notes

Acknowledgements

Funding was provided by Det Frie Forskningsråd (Grant Nos. DFF - 4002-00125, DFF - 6108-00358).

References

  1. 1.
    Alekseevsky, D.V., Cortés, V., Devchand, C.: Special complex manifolds. J. Geom. Phys. 42(1–2), 85–105 (2001)MATHGoogle Scholar
  2. 2.
    Barberis, M.L., Dotti, I.G., Verbitsky, M.: Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry. Math. Res. Lett. 16(2), 331–347 (2009)CrossRefMATHGoogle Scholar
  3. 3.
    Bryant, R. L.: Some remarks on \(\text{G}_2\)-structures. In: Akbulut, S., Önder, T., Stern, R. J. (eds.) Proceedings of the Gökova Geometry-Topology Conference 2005, pp. 75–109. International Press of Boston, Inc., Somerville, VA (2006)Google Scholar
  4. 4.
    Conti, D., Fernández, M.: Nilmanifolds with a calibrated \({{\rm G}}_2\)-structure. Differ. Geom. Appl. 29(4), 493–506 (2011)CrossRefMATHGoogle Scholar
  5. 5.
    Conti, D., Fernández, M., Santisteban, J.A.: Solvable Lie algebras are not that hypo. Transform. Groups 16(1), 51–69 (2011)CrossRefMATHGoogle Scholar
  6. 6.
    Cleyton, R., Ivanov, S.: On the geometry of closed \({{\rm G}}_2\)-structures. Commun. Math. Phys. 270(1), 53–67 (2007)CrossRefMATHGoogle Scholar
  7. 7.
    Crowley, D., Nordström, J.: New invariants of \({{\rm G}}_2\)-structures. Geom. Topol. 19(5), 2949–2992 (2015)CrossRefMATHGoogle Scholar
  8. 8.
    Freibert, M.: Cocalibrated structures on Lie algebras with a codimension one Abelian ideal. Ann. Global Anal. Geom. 42(4), 537–563 (2012)CrossRefMATHGoogle Scholar
  9. 9.
    Freibert, M.: Calibrated and parallel structures on almost Abelian Lie algebras, preprint arXiv:1307.2542 (2013)
  10. 10.
    Freibert, M., Swann, A.: Solvable groups and a shear construction. J. Geom. Phys. 106, 268–274 (2016)CrossRefMATHGoogle Scholar
  11. 11.
    Fino, A., Ugarte, L.: On generalized Gauduchon metrics. Proc. Edinb. Math. Soc. 56(3), 733–753 (2013)CrossRefMATHGoogle Scholar
  12. 12.
    Fino, A., Vezzoni, L.: Special Hermitian metrics on compact solvmanifolds. J. Geom. Phys. 91, 40–53 (2015)CrossRefMATHGoogle Scholar
  13. 13.
    Gibbons, G.W., Papadopoulos, G., Stelle, K.S.: HKT and OKT geometries on solition black hole moduli spaces. Nuclear Phys. B 508(3), 623–658 (1997)CrossRefMATHGoogle Scholar
  14. 14.
    Hitchin, N.: Stable forms and special metrics. In: Fernandez, M., Wolf, J.A., (eds.) Global Differential Geometry: The Mathematical Legacy of Alfred Gray. Contemporary Mathematics, vol. 288, pp. 70–89. American Mathematical Society (2001)Google Scholar
  15. 15.
    Ivanov, S., Papadopoulos, G.: Vanishing theorems on \((l|k)\)-strong Kähler manifolds with torsion. Adv. Math. 237, 147–164 (2013)CrossRefMATHGoogle Scholar
  16. 16.
    Mackenzie K.C.H.: Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, vol. 124. Cambridge University Press, Cambridge (1987)Google Scholar
  17. 17.
    Mackenzie, K.C.H.: General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, vol. 213. Cambridge University Press, Cambridge (2005)Google Scholar
  18. 18.
    Martín, F.: Cabrera, \({{\rm SU}}(3)\)-structures on hypersurfaces of manifolds with \({{\rm G}}_2\)-structures. Monatsh. Math. 148(1), 29–50 (2006)CrossRefGoogle Scholar
  19. 19.
    Maciá, Ó., Swann, A.: Twist geometry of the c-map. Commun. Math. Phys. 336(3), 1329–1357 (2015)CrossRefMATHGoogle Scholar
  20. 20.
    Swann, A.F.: T is for twist. In: Iglesias Ponte D. et al. (ed.) Proceedings of the XV International Workshop on Geometry and Physics, Puerto de la Cruz, September 11–16, 2006 Publicaciones de la Real Sociedad Matemática Española, vol. 11, pp. 83–94. Spanish Royal Mathematical Society (2007)Google Scholar
  21. 21.
    Swann, A.F.: Twisting Hermitian and hypercomplex geometries. Duke Math. J. 155(2), 403–431 (2010)CrossRefMATHGoogle Scholar
  22. 22.
    Swann, A.F.: Twists versus modifications. Adv. Math. 303, 611–637 (2016)CrossRefMATHGoogle Scholar
  23. 23.
    Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is t-duality. Nuclear Phys. B 479(1–2), 243–259 (1996)CrossRefMATHGoogle Scholar
  24. 24.
    Ugarte, L.: Hermitian structures on six-dimensional nilmanifolds. Transform. Groups 12(1), 175–202 (2007)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany
  2. 2.Department of MathematicsAarhus UniversityAarhusDenmark

Personalised recommendations