On characteristic foliations of metric contact-symplectic structures

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.