×

A complete classification of four-dimensional para-Kähler Lie algebras. (English) Zbl 1332.53041

In this interesting paper the author considers para-Kähler Lie algebras, that is, even-dimensional Lie algebras equipped with a pair \((J,g)\), where \(J\) is a paracomplex structure and \(g\) a pseudo-Riemannian metric, such that the fundamental 2-form \(\Omega (X,Y)=g(X,JY)\) is symplectic. A complete classification is obtained in dimension four.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C20 Global Riemannian geometry, including pinching
17B99 Lie algebras and Lie superalgebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D.V. Alekseevsky, C. Medori, A. Tomassini, Homogeneous para-Kähler Einstein manifolds, Russian Math. Surveys, 64 (2009), 1-43.; · Zbl 1179.53050
[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.; · Zbl 1165.17303
[3] P. Baird and L. Danielo, Three-dimensional Ricci solitons which project to surfaces, J. Reine Angew.Math., 608 (2007), 65-91.; · Zbl 1128.53020
[4] N. Blazić, S. Vukmirović, Four-dimensional Lie algebras with a para-hypercomplex structure, Rocky Mountain J. Math., 40 (2010), 1391-1439.; · Zbl 1207.53071
[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.; · Zbl 1264.53052
[6] G. Calvaruso, Symplectic, complex and Kähler structures on four-dimensional generalized symmetric spaces, Diff. Geom. Appl., 29 (2011), 758-769.; · Zbl 1228.53037
[7] G. Calvaruso, Four-dimensional paraKähler Lie algebras: classification and geometry, Houston J. Math., to appear.; · Zbl 1334.53018
[8] G. Calvaruso and A. Fino, Complex and paracomplex structures on homogeneous pseudo-Riemannian four-manifolds, Int. J. Math., 24 (2013), 1250130, 28 pp.; · Zbl 1266.53033
[9] G. Calvaruso and A. Fino, Ricci solitons and geometry of four-dimensional non-reductive homogeneous spaces, Canad. J. Math., 64 (2012), 778-804.; · Zbl 1252.53056
[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.; · Zbl 1405.53054
[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.; · Zbl 1201.53046
[12] V. Cruceanu, P. Fortuny and P.M. Gadea, A survey on paracomplex geometry, Rocky Mount. J. Math., 26 (1996), 83-115.; · Zbl 0856.53049
[13] B.Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Math. Soc., 197 (1974), 145-159.; · Zbl 0261.53039
[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.; · Zbl 1173.53327
[15] A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata, 7 (1978), 259-280.; · Zbl 0378.53018
[16] J. Lauret, Ricci solitons solvmanifolds, J. Reine Angew. Math., 650 (2011), 1-21.; · Zbl 1210.53051
[17] G. Ovando, Invariant complex structures on solvable real Lie groups, Manuscripta Math., 103, (2000), 19-30.; · Zbl 0972.32017
[18] G. Ovando, Four-dimensional symplectic Lie algebras, Beiträge Algebra Geom., 47(2006), no. 2, 419-434.; · Zbl 1155.53042
[19] G. Ovando, Invariant pseudo-Kähler metrics in dimension four, J. Lie Theory, 16 (2006), 371-391.; · Zbl 1102.32011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.