On characteristic foliations of metric contact-symplectic structures
Amine Hadjar
TL;DR
The paper analyzes metric contact-symplectic structures $(\eta,\omega,g)$, showing that Reeb integral curves are geodesics for any compatible metric and that all associated metrics share a closed, explicit volume form $dV$; it further proves that if the two characteristic foliations $\mathcal{S}$ and $\mathcal{C}$ are orthogonal, their leaves and the leaves of the $d\eta$-foliation $\mathcal{K}$ are minimal via Rummler’s criterion. It provides explicit constructions on nilpotent Lie groups (and their nilmanifolds) where $\mathcal{S}$ and $\mathcal{C}$ are orthogonal but not both totally geodesic, highlighting limitations of the product-type intuition. The work combines the theory of almost contact-symplectic structures with foliation geometry to establish precise geometric consequences of metric compatibility and to supply concrete model spaces. Overall, it advances understanding of how metric compatibility constraints influence Reeb dynamics, volume, and minimality of characteristic foliations in metric contact-symplectic geometry.
Abstract
We study compatible and associated metrics for a contact-symplectic pair $(η, ω)$ on a manifold. We show that the integral curves of the Reeb vector field are geodesics for any compatible metric. We prove that all associated metrics share a common volume element, which we give explicitly. When the characteristic foliations of $η$ and $ω$ are orthogonal with respect to an associated metric, their leaves, as well as those of the characteristic foliation of $dη$, are minimal. We construct explicit examples on nilpotent Lie groups and nilmanifolds where the characteristic foliations are not both totally geodesic.