# zbMATH — the first resource for mathematics

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.)
Full Text: