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

TL;DR

This work provides an explicit construction of the Fefferman metric $g^{\mathrm{F}}$ for twistor CR manifolds over a 3D conformal manifold $(\Sigma,[g])$, expressing it directly in terms of base Riemannian data and establishing a robust link between CR chains and conformal geodesics. It proves that chains and null chains on the twistor CR manifold project to conformal geodesics on $(\Sigma,[g])$, and shows that every conformal geodesic has canonical lifts to chains or null chains. A dimension-three variational framework is developed using a Kropina metric on the unit tangent sphere bundle, yielding a functional whose critical curves correspond to conformal geodesics and tying this to the total torsion functional $\mathscr{T}$. The results illuminate the interplay between CR geometry and 3D conformal geometry, provide explicit computational tools for the Fefferman metric, and offer a variational characterization with potential implications in parabolic geometry and related physical theories.

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.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1: Lee
• Definition 2.2: FKoch
• Proposition 3.1: BE
• Proposition 3.2: BE
• Proposition 3.3
• Proposition 4.1
• Lemma 4.2
• Proposition 4.3