Riemannian Curl in Contact Geometry

TL;DR

This paper introduces the contact Riemannian curl, a new differential invariant on contact manifolds with pseudo-Riemannian metrics, exploring its properties, relation to Killing forms, projective structures, and implications for the Laplace-Beltrami operator.

Contribution

It defines the contact Riemannian curl and investigates its properties, including conditions for vanishing and its relation to the Schwarzian derivative and subsymbols.

Findings

The contact Riemannian curl vanishes for constant curvature metrics with Killing 1-forms.

It depends only on the projective class of the metric, similar to the Schwarzian derivative.

The curl is proportional to the subsymbol of the Laplace-Beltrami operator and vanishes on cotangent bundles.

Abstract

We consider a contact manifold with a pseudo-Riemannian metric and define a contact vector field intrinsically associated to this pair of structures. We call this new differential invariant the contact Riemannian curl. On a Riemannian manifold, Killing vector fields are those that annihilate the metric; a Killing $1$-form is obtained from a Killing vector field by lowering indices. 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. For the Laplace-Beltrami operator on a contact manifold, the contact Riemannian curl is proportional to the subsymbol defined in arXiv:1205.6562. We also show that the contact Riemannian curl vanishes on the (co)tangent bundle over a Riemannian manifold. This implies that the corresponding subsymbol of the Laplace-Beltrami operator is identically zero.