The Fefferman metric for twistor CR manifolds and conformal geodesics in dimension three

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.