Skip to main content
SpringerLink
Log in

Hypercomplex structures with Hermitian–Norden metrics on four-dimensional Lie algebras

  • Published:
Journal of Geometry Aims and scope Submit manuscript

Abstract

Integrable hypercomplex structures with Hermitian and Norden metrics on Lie groups of dimension 4 are considered. The corresponding five types of invariant hypercomplex structures with hyper-Hermitian metric, studied by Barberis, are constructed here. The different cases regarding the signature of the basic pseudo-Riemannian metric are considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alekseevsky D.V., Marchiafava S.: Quaternionic structures on a manifold and subordinated structures. Ann. Mat. Pura Appl. CLXXI(IV): 205–273 (1996)

  2. Barberis M.L.: Hypercomplex structures on four-dimensional Lie groups. Proc. AMS 128(4), 1043–1054 (1997)

    Article  MathSciNet  Google Scholar 

  3. Barberis M.L., Dotti I.: Abelian complex structures on solvable Lie algebras. J. Lie Theory 14(1), 25–34 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Barberis M.L., Dotti I., Miatello R.: On certain locally homogeneous Clifford manifolds. Ann. Glob. Anal. Geom. 13, 289–301 (1995)

    Article  MATH  Google Scholar 

  5. Dotti I., Fino A.: Hyper-Kähler with torsion structures invariant by nilpotent Lie groups. Class. Quantum Grav. 19, 1–12 (2002)

    Article  MathSciNet  Google Scholar 

  6. Fino A., Grantcharov G.: Properties of manifolds with skew-symmetric torsion and special holonomy. Adv. Math. 189, 439–450 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ganchev G., Borisov A.: Note on the almost complex manifolds with a Norden metric. C. R. Acad. Bulgare Sci. 39, 31–34 (1986)

    MATH  MathSciNet  Google Scholar 

  8. Gray, A., Hervella, L.M.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. CXXIII(IV), 35–58 (1980)

  9. Gribachev K., Manev M.: Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure. J. Geom. 88(1–2), 41–52 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gribachev, K., Manev, M., Dimiev, S.: On the almost hypercomplex pseudo-Hermitian manifolds. In: Dimiev, S., Sekigawa, K. (eds.) Trends of Complex Analysis, Differential Geometry and Mathematical Physics, pp. 51–62. World Sci. Publ., Singapore (2003)

  11. Manev M.: A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics. J. Geom. Phys. 61(1), 248–259 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ovando G.: Invariant complex structures on solvable real Lie groups. Manuscr. Math. 103, 19–30 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Snow J.E.: Invariant complex structures on four-dimensional solvable real Lie groups. Manuscr. Math. 66, 397–412 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  14. Sommese A.: Quaternionic manifolds. Math. Ann. 212, 191–214 (1975)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Algebra and Geometry, Faculty of Mathematics, Informatics and IT, Paisii Hilendarski University of Plovdiv, 236 Bulgaria blvd, 4027, Plovdiv, Bulgaria

    Mancho Manev

Authors
  1. Mancho Manev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mancho Manev.

Additional information

This paper is partially supported by project NI13-FMI-002 of the Scientific Research Fund, Plovdiv University, Bulgaria and the German Academic Exchange Service (DAAD).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Manev, M. Hypercomplex structures with Hermitian–Norden metrics on four-dimensional Lie algebras. J. Geom. 105, 21–31 (2014). https://doi.org/10.1007/s00022-013-0188-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00022-013-0188-9

Mathematics Subject Classification (2010)

Keywords

Search

Navigation