Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Complex Manifolds

Ed. by Fino, Anna Maria

CiteScore 2017: 0.39

SCImago Journal Rank (SJR) 2017: 0.260
Source Normalized Impact per Paper (SNIP) 2017: 0.660

Mathematical Citation Quotient (MCQ) 2017: 0.18

Open Access
See all formats and pricing
More options …

A complete classification of four-dimensional paraKähler Lie algebras

Giovanni Calvaruso
  • Corresponding author
  • Dipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, Italy
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-02-09 | DOI: https://doi.org/10.1515/coma-2015-0001


We consider paraKähler Lie algebras, that is, even-dimensional Lie algebras g equipped with a pair (J, g), where J is a paracomplex structure and g a pseudo-Riemannian metric, such that the fundamental 2-form Ω(X, Y) = g(X, JY) is symplectic. A complete classification is obtained in dimension four.

Keywords: Lie algebras; paraKähler structures; pseudo-Riemannian homogeneous spaces


  • [1] D.V. Alekseevsky, C. Medori, A. Tomassini, Homogeneous para-Kähler Einstein manifolds, Russian Math. Surveys, 64 (2009), 1–43. CrossrefWeb of ScienceGoogle Scholar

  • [2] A. Andrada, M.L. Barberis, I.G. Dotti, G. Ovando, Product structures on four-dimensional solvable Lie algebras, Homology, Homotopy and Applications, 7 (2005), 9–37. Google Scholar

  • [3] P. Baird and L. Danielo, Three-dimensional Ricci solitons which project to surfaces, J. Reine Angew.Math., 608 (2007), 65–91. Web of ScienceGoogle Scholar

  • [4] N. Blazić, S. Vukmirović, Four-dimensional Lie algebras with a para-hypercomplex structure, Rocky Mountain J. Math., 40 (2010), 1391–1439. Google Scholar

  • [5] M. Brozos-Vazquez, G. Calvaruso, E. Garcia-Rio and S. Gavino-Fernandez, Three-dimensional Lorentzian homogeneous Ricci solitons, Israel J. Math., 188 (2012), 385–403. Google Scholar

  • [6] G. Calvaruso, Symplectic, complex and Kähler structures on four-dimensional generalized symmetric spaces, Diff. Geom. Appl., 29 (2011), 758–769. CrossrefGoogle Scholar

  • [7] G. Calvaruso, Four-dimensional paraKähler Lie algebras: classification and geometry, Houston J. Math., to appear. Google Scholar

  • [8] G. Calvaruso and A. Fino, Complex and paracomplex structures on homogeneous pseudo-Riemannian four-manifolds, Int. J. Math., 24 (2013), 1250130, 28 pp. CrossrefWeb of ScienceGoogle Scholar

  • [9] G. Calvaruso and A. Fino, Ricci solitons and geometry of four-dimensional non-reductive homogeneous spaces, Canad. J. Math., 64 (2012), 778–804. Google Scholar

  • [10] G. Calvaruso and A. Fino, Four-dimensional pseudo-Riemannian homogeneous Ricci soliton, Arxiv: 1111.6384. To appear in Int. J. Geom. Methods Mod. Phys. Web of ScienceGoogle Scholar

  • [11] H.-D. Cao, Recent progress on Ricci solitons, Recent advances in geometric analysis, 1–38, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2010. Google Scholar

  • [12] V. Cruceanu, P. Fortuny and P.M. Gadea, A survey on paracomplex geometry, Rocky Mount. J. Math., 26 (1996), 83–115. Google Scholar

  • [13] B.Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Math. Soc., 197 (1974), 145–159. Google Scholar

  • [14] A.S. Dancer and M.Y. Wang, Some new examples on non-Ka¨ hler Ricci solitons, Math. Res. Lett., 16 (2009), no. 2, 349–363. CrossrefGoogle Scholar

  • [15] A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata, 7 (1978), 259–280. Google Scholar

  • [16] J. Lauret, Ricci solitons solvmanifolds, J. Reine Angew. Math., 650 (2011), 1–21. Google Scholar

  • [17] G. Ovando, Invariant complex structures on solvable real Lie groups, Manuscripta Math., 103, (2000), 19–30. Google Scholar

  • [18] G. Ovando, Four-dimensional symplectic Lie algebras, Beiträge Algebra Geom., 47(2006), no. 2, 419–434. Google Scholar

  • [19] G. Ovando, Invariant pseudo-Kähler metrics in dimension four, J. Lie Theory, 16 (2006), 371–391.Google Scholar

About the article

Received: 2014-11-24

Accepted: 2015-01-10

Published Online: 2015-02-09

Citation Information: Complex Manifolds, Volume 2, Issue 1, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2015-0001.

Export Citation

© 2015 Giovanni Calvaruso. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Qi Wang, Chengming Bai, Jiefeng Liu, and Yunhe Sheng
Communications in Contemporary Mathematics, 2018
Yunhe Sheng and Rong Tang
Journal of Algebra, 2018, Volume 508, Page 256
Giovanni Calvaruso and Antonella Perrone
Differential Geometry and its Applications, 2016, Volume 45, Page 115
Giovanni Calvaruso and Anna Fino
International Journal of Geometric Methods in Modern Physics, 2015, Volume 12, Number 05, Page 1550056

Comments (0)

Please log in or register to comment.
Log in