A Product Twistor Space

∗Research supported in part by NSF grant INT-9903302.

In previous work a hyperbolic twistor space over a paraquaternionic Kähler manifold was defined, the fibre being the hyperboloid model of the hyperbolic plane with constant curvature −1. Two almost complex structures were defined on this twistor space and their properties studied. In the present paper we consider a twistor space over a paraquaternionic Kähler manifold with fibre given by the hyperboloid of 1-sheet, the anti-de-Sitter plane with constant curvature −1. This twistor space admits two natural almost product structures, more precisely almost para-Hermitian structures, which form the objects of our study.