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Anti-Kählerian manifolds. (English) Zbl 0972.53043

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.

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
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