On a criterion for local embeddability of 3-dimensional CR-structures

We introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a 3-dimensional CR-structure, which we call quasi-Fefferman metrics. These metrics generalise the Fefferman metric but allow for more control of the Ricci curvature. Our main result is a criterion for embaddability of 3-dimensional CR-structures in terms of the Ricci curvature of the quasi-Fefferman metrics in the spirit of the results by Hill et al.