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4-dimensional anti-Kähler manifolds and Weyl curvature. (English) Zbl 1157.53316

Summary: On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.

MSC:

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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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References:

[1] Arthur L. Besse: Einstein manifolds. Springer Verlag, 1987. · Zbl 0613.53001
[2] Andrzej Borowiec, Mauro Francaviglia and Igor Volvovich: Anti-Kählerian Manifolds. Differential Geometry and its Applications 12 (2000), 281–289. · Zbl 0972.53043
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