Let G be a Lie group of even dimension and let (g, J) be a left invariant anti-Kähler structure on G. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure J is abelian or bi-invariant. We find that if G admits a left invariant anti-Kähler structure (g, J) where J is abelian then the Lie algebra of G is unimodular and (G, g) is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric g for which J is an anti-isometry we obtain that the triple (G, g, J) is an anti-Kähler manifold. Besides, given a left invariant anti-Hermitian structure on G we associate a covariant 3-tensor 𝜃 on its Lie algebra and prove that such structure is anti-Kähler if and only if 𝜃 is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-Kähler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-Kähler structures).
Anti-Hermitian geometry Norden metrics B-manifolds Anti-Kähler manifold Lie groups Abelian complex structure Bi-invariant complex structure
Mathematics Subject Classification (2010)
22F30 22F50 53C50 32M10 53C55 53C15 53C56
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The authors wish to extend their sincerest appreciation and thanks to Isabel Dotti and Marcos Salvai for their corrections, comments and constructive criticisms.
Andrada, A., Barberis, M.L., Dotti, I.: Classification of abelian complex structures on 6-dimensional Lie algebras. J. Lond. Math. Soc. 83(1), 232–255 (2011)MathSciNetCrossRefMATHGoogle Scholar
Andrada, A., Barberis, M.L., Dotti, I.: Corrigendum: Classification of abelian complex structures on six-dimensional Lie algebras. J. Lond. Math. Soc. 87 (2), 319–320 (2013)MathSciNetCrossRefMATHGoogle Scholar
Borowiec, A., Ferraris, M., Francaviglia, M., Volovich, I.: Almost-complex and almost-product Einstein manifolds from a variational principle. J. Math. Phys. 40(7), 3446–3464 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
Castro, R., Hervella, L.M., García-Rio, E.: Some examples of almost complex manifolds with Norden metric. Rivista di Matematica della Università di Parma Serie 4 15, 133–141 (1989)MathSciNetMATHGoogle Scholar
Manev, M.: Classes of real isotropic hypersurfaces of a Kähler manifold with B-metric. Comptes rendus de l’Académie Bulgare des Sciences 55(4), 27–32 (2002)MATHGoogle Scholar
Mekerov, D.: Connection with parallel totally skew-symmetric torsion on almost complex manifolds with Norden metric. Comptes rendus de l’Académie Bulgare des Sciences 62(12), 1501–1508 (2009)MathSciNetMATHGoogle Scholar
Mekerov, D.: On the geometry of the connection with totally skew-symmetric torsion on almost complex manifolds with Norden metric. Comptes rendus de l’Académie Bulgare des Sciences 63(1), 19–28 (2010)MathSciNetMATHGoogle Scholar