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From Lee's paper The Fefferman Metric and Pseudo hermitian Invariants, corresponding to any 3 dimensional strictly pseudo convex CR structure, there is a conformal class of Lorentzian metrics which are not Einstein. Is there any condition on the CR structure under which a representative of the conformal class is Einstein? Or in another words, when is a Fefferman metric Einstein?

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    $\begingroup$ Lee said himself in his paper that Fefferman metric is never Einstein, but g Fefferman conformal class can have Einstein metric in local sense $\endgroup$
    – user21574
    Oct 25, 2017 at 7:40
  • $\begingroup$ You may see this paper F. Leitner: On transversally symmetric pseudo-Einstein and Fefferman-Einstein spaces, Math. Z. 256 (2007), no. 2, 443–459. $\endgroup$
    – user21574
    Oct 25, 2017 at 7:44
  • $\begingroup$ When $(M,θ,J)$ is transversally symmetric pseudo-Einstein, the conformal class of $f_θ$ contain an Einstein metric $\endgroup$
    – user21574
    Oct 25, 2017 at 7:52
  • $\begingroup$ @Hassan Jolany, How can I see that g Fefferman conformal class can have Einstein metric in local sense ? $\endgroup$
    – Masoud
    Oct 26, 2017 at 5:12
  • $\begingroup$ For $3$-dimensional CR-structures having an Einstein representative even locally forces local equivalence to the standard CR $3$-sphere. See Prop. 3 of J. Lewandowski, On the Fefferman Class of Metrics Associated with a Three-Dimensional CR Space Letters Math. Phys. 15 (1988), 129--135. $\endgroup$ Nov 2, 2017 at 18:50

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