Is the 4-dimensional manifold simply Riemannian or does it have more structure (hyperKahler, quaternionic-Kahler)? Are there any compactness assumptions?
– Pavel SafronovJun 13 '12 at 15:00

Maybe the question is about the manifolds diffeomorphic to the sphere bundle of the vector bundle of self-dual two-forms of some oriented riemannian 4-manifold ?
– BS.Jun 14 '12 at 16:44

The question is more about the six-dimensional manifold. Take some 6-dimensional manifold (say for example nearly Kaehler), when is it the twistor space of a four-dimensional manifold? Is the four-dimensional manifold then quaternionic Kaehler for example?
– MalopaJun 18 '12 at 13:54