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.


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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[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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