## 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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