Parabolic conformally symplectic structures II; parabolic contactification

Abstract

Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current article, we relate PCS-structures to parabolic contact structures. Starting from a parabolic contact structure with a transversal infinitesimal automorphism, we first construct a natural PCS-structure on any leaf space of the corresponding foliation. Then we develop a parabolic version of contactification to show that any PCS-structure can be locally realized (uniquely up to isomorphism) in this way. In the second part of the paper, these results are extended to an analogous correspondence between contact projective structures and so-called conformally Fedosov structures. The developments in this article provide the technical background for a construction of sequences and complexes of differential operators which are naturally associated to PCS-structures by pushing down BGG sequences on parabolic contact structures. This is the topic of the third part of this series of articles.