In contrast to the classical twistor spaces whose fibres are 2-spheres, we introduce twistor spaces over manifolds with almost quaternionic structures of the second kind in the sense of P. Libermann whose fibres are hyperbolic planes. We discuss two natural almost complex structures on such a twistor space and their holomorphic functions. Subject Classification (2000). 53C28, 32L25, 53C26, 53C50. MathematicsKeywords. Almost paraquaternionic structures, neutral metrics, hyperbolic twistor spaces, holomorphic functions.quaternioniennes de deuxième espèce) on a smooth manifold M . This consists of an almost complex structure J 1 and an almost product structure, J 2 such that J 1 J 2 + J 2 J 1 = 0. Setting J 3 = J 1 J 2 one has a second almost product structure which also anti-commutes with J 1 and J 2 . Now on a manifold M with such a structure, set j = y 1 J 1 + y 2 J 2 + y 3 J 3 .Then j is an almost complex structure on M if and only if −y 2 1 + y 2 2 + y 2 3 = −1 which suggests considering a hyperbolic twistor space π : Z −→ M with fibre this hyperboloid. Recall that the classical twistor space over a quaternionic Kähler manifold is a bundle over the manifold with the fibre being a 2-sphere (Salamon [18]).An almost paraquaternionic structure on a smooth manifold M is defined to be a rank 3-subbundle E of the endomorphisms bundle End(T M ) which locally is spanned by a triple {J 1 , J 2 , J 3 } which is an almost quaternionic structure of the second kind in the sense of P. Libermann.There are a number of examples of almost paraquaternionic structures including the paraquaternionic projective space as described by Blazić [5]. Under certain holonomy assumptions almost paraquaternionic structures become paraquaternionic Kähler (see e.g. Garcia-Rio, Matsushita and Vazquez-Lorenzo [9]). Even more strongly one has the notion of a neutral hyperkähler structure (see Section 4) and Kamada [14] has observed that the only compact fourmanifolds admitting such a structure are complex tori and primary Kodaira surfaces. We remark that the neutral hyperkähler four-manifolds are Ricci flat and self-dual ([14]).The tangent bundle of a differentiable manifold also carries an almost paraqua ternionic structure as studied by S. Ianus and C. Udriste [11] [12]; this includes examples where the dimension of the manifold carrying the structure is not necessarily 4n. However the most natural setting for this kind of structure is on a manifold M of dimension 4n with a neutral metric g, i.e. a pseudo-Riemannian metric of signature (2n, 2n). One reason for this is that such a metric may be given with respect to which J 1 acts as an isometry on tangent spaces and J 2 , J 3 act as anti-isometries; the effect of this is that we may define three fundamental 2-forms Ω a , a = 1, 2, 3, by Ω a (X, Y ) = g(X, J a Y ). If a neutral metric g has this property we shall say that it is adapted to the almost paraquaternionic structure E; we shall also say that J 1 , J 2 , J 3 are compatible with g. Riemannian metrics can be chosen such that g(J a X, J a Y...

The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics h t compatible with the almost complex structures J 1 and J 2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In this paper we compute the first Chern form of the almost Hermitian manifold (Z, h t , J n ), n = 1, 2 and find the geometric conditions on M under which the curvature of its Chern connection D n is of type (1, 1). We also describe the twistor spaces of constant holomorphic sectional curvature with respect to D n and show that the Nijenhuis tensor of J 2 is D 2parallel provided the base manifold M is Einstein and self-dual.