Correspondence spaces and twistor spaces for parabolic geometries

Abstract

For a semisimple Lie group $G$ with parabolic subgroups $Q\subset P\subset G$, we associate to a parabolic geometry of type $(G,P)$ on a smooth manifold $N$ the correspondence space $\Cal CN$, which is the total space of a fiber bundle over $N$ with fiber a generalized flag manifold, and construct a canonical parabolic geometry of type $(G,Q)$ on $\Cal CN$. Conversely, for a parabolic geometry of type $(G,Q)$ on a smooth manifold $M$, we construct a distribution corresponding to $P$, and find the exact conditions for its integrability. If these conditions are satisfied, then we define the twistor space $N$ as a local leaf space of the corresponding foliation. We find equivalent conditions for the existence of a parabolic geometry of type $(G,P)$ on the twistor space $N$ such that $M$ is locally isomorphic to the correspondence space $\Cal CN$, thus obtaining a complete local characterization of correspondence spaces. We show that all these constructions preserve the subclass of normal parabolic geometries (which are determined by some underlying geometric structure) and that in the regular normal case, all characterizations can be expressed in terms of the harmonic curvature of the Cartan connection, which is easier to handle. Several examples and applications are discussed.