Skip to main content
SpringerLink
Log in

On Lie Groups with Recurrent Curvature in Dimension Four

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

In this paper, we give a full classification of pseudo-Riemannian Lie groups of dimension four with non-trivial recurrent curvature tensor (i.e., non-locally symmetric). Then we investigate Ricci solitons on recurrent curvature Lie groups. Locally conformally flat examples of our classification have also been presented.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Bastami, M., Haji-Badali, A., Zaeim, A.: Recurrent curvature over four-dimensional homogeneous manifolds. RevUMA (2021). https://doi.org/10.33044/revuma.1740

  2. Calvaruso, G., Fino, A.: Four-dimensional pseudo-Riemannian homogeneous Ricci solitons. Int. J. Geom. Methods Mod. Phys. 12(5), 1550056 (2015)

    Article  MathSciNet  Google Scholar 

  3. Calvaruso, G., Zaeim, A.: Four-dimensional homogeneous Lorentzian manifolds. Monatsh. Math. 174(3), 377–402 (2013)

    Article  MathSciNet  Google Scholar 

  4. Calvaruso, G., Zaeim, A.: Four-dimensional Lorentzian Lie groups. Differ. Geom. Appl. 31(4), 496–509 (2013)

    Article  MathSciNet  Google Scholar 

  5. Calvaruso, G., Zaeim, A.: Neutral metrics on four-dimensional Lie groups. J. Lie Theory 25, 1023–1044 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Chaichi, M., Zaeim, A.: Locally homogeneous four-dimensional manifolds of signature \((2,2)\). Math. Phys. Anal. Geom. 16(3), 345–361 (2013)

    Article  MathSciNet  Google Scholar 

  7. Dubey, R.S.D.: Generalized recurrent spaces. Indian J. Pure Appl. Math. 10, 1508–1513 (1979)

    MathSciNet  MATH  Google Scholar 

  8. García-Río, E., Gilkey, P., Nikčević, S.: Homogeneity of Lorentzian three manifolds with recurrent curvature. Math. Nachr. 287, 32–47 (2014)

    Article  MathSciNet  Google Scholar 

  9. Gilkey, P.: Geometric Properties of neutral Operators Defined by the Riemannian curvature tensor. World Scientific Publishing Co. Inc., River Edge (2001)

    Book  Google Scholar 

  10. Haji-Badali, A., Karami, R.: Ricci soliton on four-dimensional neutral Lie group. J. Lie Theory 27, 943–967 (2017)

    MathSciNet  MATH  Google Scholar 

  11. Haji-Badali, A., Zaeim, A.: Semi-symmetric four dimensional neutral Lie groups. Czech Math. J. 70, 393–410 (2020)

    Article  MathSciNet  Google Scholar 

  12. Haji-Badali, A.: Ricci almost soliton on three-dimensional manifolds with recurrent curvature. Mediterr. J. Math. (2017). https://doi.org/10.1007/s00009-016-0810-9

    Article  MathSciNet  MATH  Google Scholar 

  13. Karami, R., Zaeim, A., Haji-Badali, A.: Ricci solitons and geometry of four dimensional Einstein-like neutral Lie groups. Period Math. Hung. 78, 58–78 (2019)

    Article  MathSciNet  Google Scholar 

  14. Komrakov Jr., B.: Einstein–Maxwell equation on four-dimensional homogeneous spaces. Lobachevskii J. Math. 8, 33–165 (2001)

    MathSciNet  MATH  Google Scholar 

  15. Mantica, C.A., Molinari, L.G.: On conformally recurrent manifolds of dimension greater than 4. Int. J. Geom. Meth. Mod. Phys. 13, 17–26 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Mathur, S.B.: on recurrent space. Proc. Natl. Inst. Sci. India 34, 40–49 (1966)

    MathSciNet  Google Scholar 

  17. Olszak, K., Olszak, Z.: on pseudo-Riemannian manifolds with recurrent concircular curvature tensor. Acta Math. Hung. 137, 64–71 (2012)

    Article  MathSciNet  Google Scholar 

  18. Ruse, H.S.: Three-dimensional space of recurrent curvature. Proc. Lond. Math. Soc. S2–50(1), 438–446 (1948)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Basic Sciences Faculty, University of Bonab, Bonab, 5551761167, Iran

    Milad Bastami & Ali Haji-Badali

  2. Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran

    Amirhesam Zaeim

Authors
  1. Milad Bastami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amirhesam Zaeim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ali Haji-Badali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali Haji-Badali.

Additional information

Communicated by Mohammad Koushesh.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bastami, M., Zaeim, A. & Haji-Badali, A. On Lie Groups with Recurrent Curvature in Dimension Four. Bull. Iran. Math. Soc. 48, 1037–1049 (2022). https://doi.org/10.1007/s41980-021-00561-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-021-00561-3

Keywords

Mathematics Subject Classification

Search

Navigation