Special Hermitian metrics and Lie groups. (English) Zbl 1247.32024
Summary: A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form \(\omega \) is \(\partial \bar{\partial }\)-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a \(2n\)-dimensional SKT Lie algebra \(g\) and using a Hermitian flat connection on \(g\) we construct a \(4n\)-dimensional SKT Lie algebra. We apply this method to some 4-dimensional SKT Lie algebras. Moreover, we classify symplectic forms taming complex structures on 4-dimensional Lie algebras.
MSC:
| 32Q15 | Kähler manifolds |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 53C30 | Differential geometry of homogeneous manifolds |
| 53D05 | Symplectic manifolds (general theory) |
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