Geometry of neutral metrics in dimesnion four

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The purpose of this article is to review some recent results on the geometry of neutral signature metrics in dimension four and their twistor spaces. The following topics are considered: Neutral Kähler and hyperkähler surfaces, Walker metrics, Neutral anti-self-dual 4-manifolds and projective structures, Twistor spaces of neutral metrics.
20 pages, invited talk at the Thirty Seventh Spring Connference of the Union of Bulgarian Mathematicians, Borovets, April 2-6, 2008

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