Holomorphic isometries of twistor spaces

Abstract

Let Z g ( M) be the twistor space over an oriented 2 n-dimensional Riemannian manifold ( M, g) with nonpositive and parallel Ricci tensor. Let h and J be the natural metric and almost complex structure on Z g ( M), respectively. We prove that any isometry of the twistor space Z g ( M) preserves the horizontal and vertical distributions. When M is compact, we give an estimate of the dimension of the groups of isometries and holomorphic isometries on the twistor space. In particular, if M is a compact almost Kähler manifold with the same properties of curvature and the Ricci tensor is negative, then the group of the holomorphic isometries is finite.