Contact Symplectic Fibrations and Bundle Contact Structures

We consider a certain type of fiber bundles with odd dimensional connected contact base, exact symplectic fibers, and the structure group contained in the group of exact symplectomorphisms of the fiber. We call such fibrations "contact symplectic fibrations". By a result of Hajduk-Walczak, some of these admit contact structures which are "compatible" (in a certain sense) with the corresponding fibration structures. We show that this result can be extended to get a compatible contact structure, called a "bundle contact structure", on the total space of any compact contact symplectic fibration. We also verify some flexibility results for bundle contact structures: The isotopic contact structures on the base or variations of the exact symplectic structure on the fiber produce isotopic bundle contact structures on the total space. Moreover, we also discuss some naturality results of such fibrations: We prove that any smooth embedding of a section into a compact contact symplectic fibration equipped with a bundle contact structure is, indeed, a contact embedding, and also show that any two such contact sections are contactomorphic.