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On Bochner flat para-Kählerian manifolds. (English) Zbl 1107.53021
Summary: Let \(B\) be the Bochner curvature tensor of a para-Kählerian manifold. It is proved that if the manifold is Bochner parallel \((\nabla B = 0)\), then it is Bochner flat \((B = 0)\) or locally symmetric \((\nabla R = 0)\). Moreover, we define the notion of the paraholomorphic pseudosymmetry of a para-Kählerian manifold. We find necessary and sufficient conditions for a Bochner flat para-Kählerian manifold to be paraholomorphically pseudosymmetric. Especially, in the case when the Ricci operator is diagonalizable, a Bochner flat para-Kählerian manifold is paraholomorphically pseudosymmetric if and only if the Ricci operator has at most two eigenvalues. A class of examples of manifolds of this kind is presented.
MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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