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On \(HB\)-recurrent hyperbolic Kaehlerian spaces. (English) Zbl 0827.53018
By a hyperbolic Kählerian space the author means a triple \((M, F, g)\), where \(M\) is a \(2m\)-dimensional differentiable manifold, \(F\) is a (1,1)- tensor field and \(g\) is a pseudo-Riemannian metric on \(M\) such that \(\nabla F = 0\), \(F^2 = \text{Id}\) and \(g(FX,Y) = -g(X,FY)\) for any vector fields \(X\), \(Y\) on \(M\). Such manifolds are also called to be para- Kählerian [cf. P. Libermann, Ann. Mat. Pura Appl., IV. Ser. 36, 27-120 (1954; Zbl 0056.154)]. In her previous paper, the author defined the so-called \(HB\)-tensor on a hyperbolic Kählerian space, which is an analogy of the famous Bochner curvature tensor of Kählerian manifolds. In the present paper, certain results on hyperbolic Kählerian spaces with recurrent \(HB\)-tensor \((\nabla HB = \kappa \otimes HB)\) and recurrent Ricci tensor \((\nabla \text{Ric} = \kappa^* \otimes \text{Ric})\) are proved. For instance, it is shown that if \(\dim M > 4\), then both the recurrence forms \(\kappa\), \(\kappa^*\) are closed. Moreover, such assumptions can always be reduced to certain weaker conditions.

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