Twistor spaces and six-dimensional manifolds

What are the conditions on a six-dimensional manifold to be the twistor space of a four-dimensional one?

flag	asked Jun 13 at 14:16 Malopa 21●1

	Is the 4-dimensional manifold simply Riemannian or does it have more structure (hyperKahler, quaternionic-Kahler)? Are there any compactness assumptions? – Pavel Safronov Jun 13 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 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? – Malopa Jun 18 at 13:54