Anti-Kählerian manifolds. (English) Zbl 0972.53043

Borowiec, Andrzej; Francaviglia, Mauro; Volovich, Igor

Differ. Geom. Appl. 12, No. 3, 281-289 (2000).

An anti-Kählerian manifold is a smooth manifold \(M\), an almost complex structure \(J\) and a metric \(g\) such that \(J\) is annihilated by the Levi-Civita connection of \(g\) (see thus the analogy to Kähler manifolds) and \(g\) is anti-Hermitian, i.e., \(g(JX,JY)=-g(X,Y)\) for all vector fields \(X\) and \(Y\) on \(M\). For such manifolds, the authors show that \(g\) is the real part of a certain holomorphic metric on \(M\) and that the odd Chern numbers of \(M\) must vanish. Complex parallelizable manifolds are anti-Kählerian. A method of generating new solutions of Einstein equations by means of the complexification of a given Einstein metric is also presented.

Reviewer: Eugen Pascu (Montreal)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C56	Other complex differential geometry
53C80	Applications of global differential geometry to the sciences

Keywords:

complex manifold; anti-Hermitian metric; parallel almost complex structure; Einstein metric; holomorphic metric; complex Riemannian geometry; Kähler manifold