2025/02/13 by Katz, Gabriel
#Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.09773
Let β be a contact form on a compact smooth manifold X and vβ its Reeb vector field. The paper applies general results of different authors about Hodge structures that are transversal to a given foliation to the special case of 1-dimensional foliation generated by the Reeb flow vβ. Theses applications are available for the Reeb flows on \sf closed manifolds X. In contrast, for the Reeb flows on manifolds with boundary, little is known about the Hodge structures transversal to the vβ-flow. We are trying to fill in this gap.\smallskip The de Rham differential complex Ω\mathsf b^∗(X, vβ) of, so called, \sf basic relative to vβ-flow differential forms is in the focus of this investigation. By definition, the basic forms vanish when being contracted with vβ, and so do their differentials. In particular, we investigate when the 2-form dβ and its powers deliver nontrivial elements in the basic de Rham cohomology H^∗basic dR(X, vβ) of the differential complex Ω\mathsf b^∗(X, vβ). Answers to these questions seem to contrast sharply the cases of a closed X and a X with boundary. %we prove that when a vβ-flow admits a Lyapunov function, then the basic de Rham cohomology H^∗basic dR(X, vβ) of the complex Ω\mathsf b^∗(X, vβ) are topological invariants of X. On the other hand, building on work of Raźny \citeRaz, we show that on closed manifolds, equipped with a transversal to the Reeb flow Hodge structure that satisfies the \sf Basic Hard Lefschetz property, the basic de Rham cohomology H^∗basic dR(X, vβ) are topological invariants of X.