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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.

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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