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On Bochner semisymmetric para-Kählerian manifolds. (English) Zbl 1029.53038
Starting with the fact that a semisymmetric (\(R\cdot R=0\)) para-Kählerian manifold is Bochner semisymmetric (\(R\cdot B=0\)), the author proves a partial inverse theorem: If a para-Kählerian manifold \(M\) is Bochner semisymmetric and the Bochner curvature tensor \(B\) does not vanish at each points \(x\in M\), then \(M\) is semisymmetric. Moreover, the author provides examples of para-Kählerian manifolds which are: (1) semisymmetric and not Bochner flat; (2) Bochner flat and not semisymmetric.

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics