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.