A criterion for local embeddability of three-dimensional CR structures

  • Gerd SchmalzEmail author
  • Masoud Ganji


We introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a three-dimensional CR structure, which we call FRT metrics. These metrics generalise the Fefferman metric, allowing for more control of the Ricci curvature, but are more special than the shearfree Lorentzian metrics introduced by Robinson and Trautman.Our main result is a criterion for embeddability of three-dimensional CR structures in terms of the Ricci curvature of the FRT metrics in the spirit of the results by Lewandowski et al. (Class Quantum Gravity 7(11):L241–L246, 1990) and also Hill et al. (Indiana Univ Math J 57(7):3131–3176, 2008.


Embeddability of CR manifolds Shearfree null congruence Lorentzian manifolds 

Mathematics Subject Classification

32V30 53B30 83C05 



The authors wish to express their gratitude to Howard Jacobowitz and Jerzy Lewandowski for inspiring and useful discussions.


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Science and TechnologyUniversity of New EnglandArmidaleAustralia

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