Contact de Rham Cohomology and Hodge Structures Transversal to Reeb Foliations

Let β be a contact form on a compact smooth manifold X and vβ its Reeb vector field. This study applies the general results of different authors regarding Hodge structures that are transversal to a given foliation to the special case of 1-dimensional foliation generated by the Reeb flow vβ. The de Rham differential complex Ωbasic*(X,vβ) of so-called basic forms relative to vβ-flow differential forms is the focus of this investigation. By definition, basic forms vanish when being contracted with vβ, and so do their differentials. We prove that under the change of β⇝β1=β+df, where a function f:X→R such that df(vβ)>−1, the differential complexes Ωbasic*(X,vβ1) and Ωbasic*(X,vβ) are canonically isomorphic. We investigate when the 2-form dβ and its powers deliver nontrivial elements in the basic de Rham cohomology HbasicdR*(X,vβ) of the differential complex Ωbasic*(X,vβ). Answers to these questions contrast sharply in the cases of a closed X and an X with boundary. Building on the work of Raźny, we show that on a closed manifold X equipped with a transversal to the Reeb flow Hodge structure that satisfies the Basic Hard Lefschetz Property, the basic de Rham cohomology HbasicdR*(X,vβ) is a topological invariant of X.