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Correspondence spaces and twistor spaces for parabolic geometries. (English) Zbl 1075.53022
In this paper, a semisimple Lie group $$G$$ and parabolic subgroups $$Q\subset P\subset G$$ are considered. A parabolic geometry of type $$(G, P)$$ on a manifold $$N$$ is associated with the fiber bundle $$\mathcal CN$$ over $$N$$ such that its fiber is a generalized flag manifold. A canonical parabolic geometry of type $$(G, Q)$$ on $$\mathcal CN$$ is constructed. Conversely, for a parabolic geometry of type $$(G, Q)$$ on a manifold $$M$$, a distribution corresponding to $$P$$ is constructed. The integrability conditions of this distribution are found. The twistor space $$N$$ as a local leaf space of the corresponding foliation is defined. The existing conditions of a parabolic geometry of type $$(G, P)$$ on $$N$$ such that $$M$$ is locally isomorphic to the space $$\mathcal CN$$ are found. It is shown that all these constructions preserve the subclass of normal parabolic geometries and all characterizations can be expressed in terms of the harmonic curvature of the Cartan connection in a regular normal case. Several examples and applications are discussed

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C28 Twistor methods in differential geometry
##### Keywords:
parabolic subgroup; parabolic geometry; twistor space
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