The Fefferman Metric for Twistor CR Manifolds and Conformal Geodesics in Dimension Three

TL;DR

This paper explicitly describes the Fefferman metric for twistor CR manifolds in three dimensions and explores the relationship between chains, null chains, and conformal geodesics, providing a variational characterization.

Contribution

It provides an explicit Riemannian description of the Fefferman metric for twistor CR manifolds and links chains to conformal geodesics in dimension three.

Findings

Chains and null chains project to conformal geodesics.

Any conformal geodesic lifts to a chain and a null chain.

A variational principle involving total torsion characterizes conformal geodesics.

Abstract

We give an explicit description of the Fefferman metric for twistor CR manifolds in terms of Riemannian structures on the base conformal 3-manifolds. As an application, we prove that chains and null chains on twistor CR manifolds project to conformal geodesics, and that any conformal geodesic has lifts both to a chain and a null chain. By using this correspondence, we give a variational characterization of conformal geodesics in dimension three which involves the total torsion functional.