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

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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 '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? –  Malopa Jun 18 '12 at 13:54

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