On characteristic foliations of metric contact-symplectic structures

TL;DR

This paper investigates the geometric properties of characteristic foliations in metric contact-symplectic structures, revealing conditions for geodesic and minimal foliations, and providing explicit examples on nilpotent Lie groups.

Contribution

It demonstrates that Reeb vector field integral curves are geodesics for compatible metrics, shows all associated metrics share a volume element, and constructs examples with non-totally geodesic foliations.

Findings

Reeb vector field curves are geodesics for compatible metrics

All associated metrics share a common volume element

Characteristic foliations can be non-totally geodesic in explicit examples

Abstract

We study compatible and associated metrics for a contact-symplectic pair $(\eta , \omega)$ 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 $\eta$ and $\omega$ are orthogonal with respect to an associated metric, their leaves, as well as those of the characteristic foliation of $d\eta$, are minimal. We construct explicit examples on nilpotent Lie groups and nilmanifolds where the characteristic foliations are not both totally geodesic.