Einstein Metrics Adapted to Contact Structures on 3-Manifolds

TL;DR

This paper explores the relationship between Einstein metrics and contact structures on 3-manifolds, establishing conditions for adapted metrics and analyzing special cases like hyperbolic, flat, and elliptic geometries.

Contribution

It characterizes when a contact 3-manifold admits an adapted metric, linking geometric structures with Einstein metrics and solving the field equations locally.

Findings

Hyperbolic space forms lack adapted contact structures.

Contact structures on flat or elliptic space forms are contact isometric to the standard.

The torsion is constant in Einstein adapted contact structures.

Abstract

The Newman-Penrose-Perjes formalism is applied to smooth contact structures on riemannian 3-manifolds. In particular it is shown that a contact 3-manifold admits an adapted riemannian metric if and only if it admits a metric with a divergence-free, constantly twisting, geodesic congruence. The shear of this congruence is identified with the torsion of the associated pseudohermitian structure, while the Tanaka-Webster curvature is identified with certain derivatives of the spin coefficients. The particular case where the associated riemannian metric is Einstein is studied in detail. It is found that the torsion is constant and the field equations are completely solved locally. Hyperbolic space forms are shown not to have adapted contact structures, even locally, while contact structures adapted to a flat or elliptic space form are contact isometric to the standard one.