Holomorphic isometries of twistor spaces (Q1602453)

The twistor space of a smooth \(2n\)-dimensional Riemannian manifold \(M\) is the natural \(SO(2n)/\) \(U(n)\)-bundle \(Z(M)\) over \(M\). The Levi-Civita connection of \(M\) determines a connection and hence a natural Riemannian metric on \(Z(M)\) in such a way that the natural projection \(Z(M)\to M\) is a Riemannian submersion. The authors show that if the Ricci-tensor of \(M\) is parallel and non-positive then every isometry of \(Z(M)\) preserves the horizontal and vertical distributions. If in addition the Ricci tensor is negative then the Bochner formula shows that there are no nontrivial Killing fields on \(M\). As a consequence, in this case \(Z(M)\) does not admit a Killing field which is holomorphic with respect to the natural almost complex structure.