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
This is a preview of subscription content, log in to check access.
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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)zbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar