Riemannian curl in contact geometry

We consider a contact manifold with a pseudo-Riemannian metric and define a contact vector field associated with this pair of structures. We call this new differential invariant the contact Riemannian curl. We show that the contact Riemannian curl vanishes if the metric is of constant curvature and the contact structure is defined by a Killing 1-form. We also show that the contact Riemannian curl has a strong similarity with the Schwarzian derivative, since it depends only on the projective equivalence class of the metric. The subsymbol of the Laplace-Beltrami operator corresponding to a metric on a contact manifold is proportional to the contact Riemannian curl. We also show that the contact Riemannian curl vanishes on the spherical (co)tangent bundle over a Riemannian manifold.