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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio


IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

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1572-9176
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On the geometrical properties of hypercomplex four-dimensional Lie groups

Mehri Nasehi / Mansour Aghasi
  • Corresponding author
  • Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran
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Published Online: 2018-02-07 | DOI: https://doi.org/10.1515/gmj-2018-0003

Abstract

In this paper, we first classify Einstein-like metrics on hypercomplex four-dimensional Lie groups. Then we obtain the exact form of all harmonic maps on these spaces. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and determine all critical points for their energy functional restricted to vector fields of the same length. Furthermore, we give a complete and explicit description of all totally geodesic hypersurfaces of these spaces. The existence of Einstein hypercomplex four-dimensional Lie groups and the non-existence of non-trivial left-invariant Ricci and Yamabe solitons on these spaces are also proved.

Keywords: Harmonicity of invariant vector fields; conformal flatness; Ricci solitons; totally geodesic hypersurfaces; Yamabe solitons

MSC 2010: 53C30; 53C15

References

  • [1]

    M. Aghasi and M. Nasehi, Some geometrical properties of a five-dimensional solvable Lie group, Differ. Geom. Dyn. Syst. 15 (2013), 1–12. Google Scholar

  • [2]

    M. Aghasi and M. Nasehi, On the geometrical properties of solvable Lie groups, Adv. Geom. 15 (2015), no. 4, 507–517. Web of ScienceGoogle Scholar

  • [3]

    M. L. Barberis, Hypercomplex structures on four-dimensional Lie groups, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1043–1054. CrossrefGoogle Scholar

  • [4]

    M. L. Barberis, Hyper-Kähler metrics conformal to left invariant metrics on four-dimensional Lie groups, Math. Phys. Anal. Geom. 6 (2003), no. 1, 1–8. CrossrefGoogle Scholar

  • [5]

    G. Calvaruso, Harmonicity properties of invariant vector fields on three-dimensional Lorentzian Lie groups, J. Geom. Phys. 61 (2011), no. 2, 498–515. CrossrefWeb of ScienceGoogle Scholar

  • [6]

    G. Calvaruso, Harmonicity of vector fields on four-dimensional generalized symmetric spaces, Cent. Eur. J. Math. 10 (2012), no. 2, 411–425. Web of ScienceCrossrefGoogle Scholar

  • [7]

    G. Calvaruso and J. Van der Veken, Totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups, Results Math. 64 (2013), no. 1–2, 135–153. Web of ScienceCrossrefGoogle Scholar

  • [8]

    G. Calvaruso and A. Zaeim, A complete classification of Ricci and Yamabe solitons of non-reductive homogeneous 4-spaces, J. Geom. Phys. 80 (2014), 15–25. Web of ScienceCrossrefGoogle Scholar

  • [9]

    E. Calviño Louzao, J. Seoane-Bascoy, M. E. Vázquez-Abal and R. Vázquez-Lorenzo, Three-dimensional homogeneous Lorentzian Yamabe solitons, Abh. Math. Semin. Univ. Hambg. 82 (2012), no. 2, 193–203. CrossrefGoogle Scholar

  • [10]

    B. De Leo and J. Van der Veken, Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces, Geom. Dedicata 159 (2012), 373–387. CrossrefGoogle Scholar

  • [11]

    G. W. Gibbons, G. Papadopoulos and K. S. Stelle, HKT and OKT geometries on soliton black hole moduli spaces, Nuclear Phys. B 508 (1997), no. 3, 623–658. CrossrefGoogle Scholar

  • [12]

    A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata 7 (1978), no. 3, 259–280. Google Scholar

  • [13]

    H. R. S. Moghaddam, Randers metrics of Berwald type on four-dimensional hypercomplex Lie groups, J. Phys. A: Math. Theor. 42 (2009), no. 9, Article ID 095212. Web of ScienceGoogle Scholar

  • [14]

    D. Perrone, Almost contact metric manifolds whose Reeb vector field is a harmonic section, Acta Math. Hungar. 138 (2013), no. 1–2, 102–126. CrossrefGoogle Scholar

  • [15]

    Y. S. Poon, Examples of hyper-Kähler connections with torsion, Quaternionic Structures in Mathematics and Physics (Rome 1999), Universitá Studi Roma “La Sapienza”, Rome (1999), 321–327. Google Scholar

  • [16]

    H. R. Salimi Moghaddam, On some hypercomplex 4-dimensional Lie groups of constant scalar curvature, Int. J. Geom. Methods Mod. Phys. 6 (2009), no. 4, 619–624. CrossrefGoogle Scholar

About the article

Received: 2015-04-18

Revised: 2016-05-25

Accepted: 2016-10-24

Published Online: 2018-02-07


Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2018-0003.

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