Twistor Spaces of Non-flat Bochner-kähler Manifolds

This can be compared to the fact that a conformally flat manifold of dimension n > 2 is locally conformal to a region of the sphere of the same dimension. In twistor theory, it is well-known that an even dimensional conformally flat manifold has an integrable twistor space ([7], [6], [1], [3]). It is interesting, as an analogy, that a Bochner-Kähler manifold has integrable twistor spaces defined by O’Brian and Rawnsley in [7]. Since it was proved by calculating the Nijenhuis tensors, the proof does not give holomorphic coordinates. Hence it would be natural to ask to “give a system of local coordinates of the twistor spaces of the above model spaces”. Furthermore, to study a generalization of the Penrose transform, it is necessary to “construct the moduli space of relative deformations of fibers”, which we call the complexification. In the present paper, we give answers to the above two questions. Since the first space is flat, its twistor space as a Riemannian manifold has already an integrable complex structure. Hence the problems can be solved easily by restricting the Penrose diagram as a Riemannian manifold. In the second case, we give local coordinates by constructing a local embedding to a product of complex projective spaces (Theorem 3.1 and Theorem 3.3). The complexification of H × P is H × H̄ ×