A product twistor space

Abstract. In previous work a hyperbolic twistor space over a paraquater-nionic 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. 1. Introduction and hyperbolic twistor spaces. In [2, 3] we introduced a hyperbolic twistor space which we will need to review. The starting point was the following simple observation. In [12] P. Libermann introduced the notion of an almost quaternionic structure of the second kind (presque quater