Geodesic
Invariant Pseudo-K�hler Metrics in Dimension Four
Journal of Lie theory, Tome 16 (2006) no. 2, pp. 371-391.

Voir la notice de l'article provenant de la source Heldermann Verlag

Four dimensional simply connected Lie groups admitting a pseudo K�hler metric are determined. The corresponding Lie algebras are modelled and the compatible pairs (J, ω) are parametrized up to complex isomorphism (where J is a complex structure and ω is a symplectic structure). Such structure gives rise to a pseudo-Riemannian metric g, for which J is a parallel. It is proved that most of these complex homogeneous spaces admit a compatible pseudo-K�hler Einstein metric. Ricci flat and flat metrics are determined. In particular Ricci flat unimodular pseudo-K�hler Lie groups are flat in dimension four. Other algebraic and geometric features are treated. A general construction of Ricci flat pseudo-K�hler structures in higher dimension on some affine Lie algebras is given. Walker and hypersymplectic metrics are compared.
Classification : 32Q15, 32Q20, 53C55, 32M10, 57S25, 22E25
Mots-clés : Pseudo-Kaehler metrics, Kaehler Lie algebras, invariant metrics, four dimensional Lie algebras 
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     author = {G. P. Ovando },
     title = {Invariant {Pseudo-K�hler} {Metrics} in {Dimension} {Four}},
     journal = {Journal of Lie theory},
     pages = {371--391},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2006},
     url = {http://geodesic.mathdoc.fr/item/JLT_2006_16_2_JLT_2006_16_2_a9/}
}
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G. P. Ovando . Invariant Pseudo-K�hler Metrics in Dimension Four. Journal of Lie theory, Tome 16 (2006) no. 2, pp. 371-391. http://geodesic.mathdoc.fr/item/JLT_2006_16_2_JLT_2006_16_2_a9/