Contact de Rham Cohomology and Hodge Structures Transversal to Reeb Foliations

Let β be a contact form on a compact smooth manifold X and v β 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 v β . The de Rham differential complex Ω basic * ( X , v β ) of so-called basic forms relative to v β -flow differential forms is the focus of this investigation. By definition, basic forms vanish when being contracted with v β , and so do their differentials. We prove that under the change of β ⇝ β 1 = β + d f , where a function f : X → R such that d f ( v β ) > − 1 , the differential complexes Ω basic * ( X , v β 1 ) and Ω basic * ( X , v β ) are canonically isomorphic. We investigate when the 2-form d β and its powers deliver nontrivial elements in the basic de Rham cohomology H basic d R * ( X , v β ) of the differential complex Ω basic * ( X , v β ) . 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 H basic d R * ( X , v β ) is a topological invariant of X .