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On HB-flat hyperbolic Kaehlerian spaces. (English) Zbl 0964.53017
The author studies one special class of pseudo-Riemannian spaces, namely hyperbolic Kählerian spaces. Since, in a hyperbolic Kählerian space, a conformal transformation cannot be naturally introduced [Zb. Rad. Filoz. Fak. Nišu, Ser. Mat. 4, 55-64 (1990; Zbl 0707.53023)], the author investigated a conformal connection and found a tensor, named HB-tensor, which is an invariant for all conformal connections on a hyperbolic Kählerian space. Here, the author investigates a hyperbolic Kählerian space with vanishing HB-tensor and proves a theorem which shows full analogy between conformally flat Riemannian spaces, Bochner-flat Kählerian spaces and HB-flat hyperbolic Kählerian spaces.
Moreover, the author proves a theorem asserting that a hyperbolic HB-flat Kählerian space is an Einstein space if and only if it is a space of almost constant curvature. Furthermore, the author proves a theorem connected with HB-flat decomposable hyperbolic Kählerian spaces, corresponding to a similar one in the Kähler case.
In order to prove these results, the author constructs effectively a special basis, called a separated basis, in which a hyperbolic Kählerian space is divided very naturally into two totally geodesic subspaces of equal dimension.

MSC:
53B35 Local differential geometry of Hermitian and Kählerian structures
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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