A criterion for local embeddability of three-dimensional CR structures
type: Item
instance of: scholarly article
title: A criterion for local embeddability of three-dimensional CR structures
zbMATH Open document ID: 1423.32031
author: Gerd Schmalz
author: M. Ganji
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2019-03-26T00:00:00Z
review text: Let \((M, D, J)\) be a \(3\)-dimensional CR manifold with a contact structure \(D\) and a smooth field of endomorphisms \(J\) such that \(J^2 = -\mathrm{id}\). The local embeddability problem asks whether there exists a local embedding \(\iota: M \hookrightarrow\mathbb{C}^2\) such that the CR structures \(\iota M\) induced by \(\iota\) and by \(\mathbb{C}^2\) coincide, and this problem is equivalent to finding two functionally independent CR functions \(f\) satisfying \(\bar \partial f =0\). It turns out [\textit{J.-P. Rosay}, Ann. Inst. Fourier 39, No. 3, 811--823 (1989; Zbl 0674.32008)] that the existence of nonconstant solutions to this equation is not sufficient to guarantee embeddability of the CR structure. A theorem by \textit{H. Jacobowitz} [Pac. J. Math. 127, No. 1, 91--101 (1987; Zbl 0583.32050)] provides a criterion for embeddability of CR structures in terms of the canonical bundle. In this paper, the authors introduce a family of Lorentzian metrics on a circle bundle over the CR manifold \(M\) which is more general than the conformal class of Fefferman metrics but more special than the family of Lorentzian metrics that admits a shearfree congruence, introduced by \textit{A. Trautman} [Classical Quantum Gravity 19, No. 2, R1--R10 (2002; Zbl 1008.53054)]. The authors call these metrics FRT metrics for Fefferman, Robinson and Trautman. The authors prove that a pseudoconvex CR structure \((M, D, J)\) is locally embeddable if and only if it amits an FRT metric on a circle bundle over \(M\) for which the complexified Ricci curvature vanishes on the \(\alpha\)-planes given by the \(\sqrt{-1}\) eigenspace of \(J\).
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 32V30
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 53B30
zbMATH DE Number: 7041961
zbMATH Keywords: CR structure
zbMATH Keywords: embeddability of CR manifolds
zbMATH Keywords: shearfree null congruence
zbMATH Keywords: Lorentzian metrics
MaRDI profile type: Publication
full work available at URL: s10231-018-0785-1
OpenAlex ID: W2886549580
cites work: Sharp version of the Goldberg-Sachs theorem
cites work: CR-tractors and the Fefferman space
cites work: Einstein's equations and the embedding of 3-dimensional CR manifolds
cites work: On transversally symmetric pseudo-Einstein and Fefferman-Einstein spaces
cites work: Q5721035
cites work: The canonical bundle and realizable CR hypersurfaces
cites work: Q3998484
cites work: Non-realizable CR structures
cites work: New examples of non-locally embeddable CR structures (with no non- constant CR distributions)
cites work: On the Fefferman class of metrics associated with a three-dimensional CR space
cites work: Einstein's equations and realizability of CR manifolds
cites work: Introduction to Smooth Manifolds
cites work: Q4063860
cites work: Robinson manifolds and Cauchy–Riemann spaces
Wikidata QID: Q129409062
DOI: 10.1007/S10231-018-0785-1
description: scientific article; zbMATH DE number 7041961