Perturbations of Fefferman spaces over almost CR manifolds

1. Introduction

2. Almost CR geometry

3. Conformal geometry

4. The Fefferman construction

5. Perturbed Fefferman spaces

6. Distinguished almost Lorentzian scales

7. Properties of the zero set of almost (half-)Einstein scales

8. Comments on conformal symmetries

Appendix A An almost CR manifold admitting a CR–Einstein structure

Appendix B Proof of Theorem 4.6

References

2.1. General definitions

2.2. Webster connections

2.3. Transformation rules

2.4. Gauged Webster connections

2.5. CR–Einstein structures

3.1. Optical geometry

3.2. Almost Robinson geometry

3.3. Twist-induced nearly Robinson geometry with non-shearing congruence

3.4. Almost Einstein scales and generalisations

4.1. Fefferman spaces

4.2. Characterisations

5.1. Twisting non-shearing congruences of null geodesics

5.2. Curvature conditions

6.1. Almost weakly half-Einstein scales

6.2. Almost half-Einstein scales

6.3. Almost Einstein scales

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

Definition 2.1.

Lemma 2.2.

Proof.

Remark 2.3.

Theorem 2.4.

Proof.

Definition 2.5.

Remark 2.6.

Proposition 2.7.

Theorem 2.8.

Proof.

Remark 2.9.

Proposition 2.10.

Proof.

Remark 2.11.

Theorem 3.1 ([TC22]).

Remark 3.2.

Definition 3.3.

Definition 3.4.

Remark 3.5.

Definition 3.6.

Definition 3.7.

Definition 3.8.

Remark 3.9.

Proposition 3.10.

Proof.

Proposition 3.11.

Proof.

Definition 4.1.

Definition 4.2.

Remark 4.3.

Remark 4.4.

Theorem 4.5 (\citesGraham1987,Cap2008).

Theorem 4.6.

Remark 4.7.

Remark 4.8.

Definition 5.1.

Lemma 5.2.

Proof.

Definition 5.3.

Example 5.4.

Remark 5.5.

Lemma 5.6.

Lemma 5.7.

Proof.

Proposition 5.8.

Proof.

Remark 5.9.

Remark 5.10.

Proposition 5.11.

Proof.

Remark 5.12.

Remark 5.13.

Conjecture 5.14.

Proposition 6.1.

Proof.

Remark 6.2.

Corollary 6.3.

Proposition 6.4.

Remark 6.5.

Lemma 6.6.

Proof.

Lemma 6.7.

Remark 6.8.

Proposition 6.9.

Proposition 6.10.

Corollary 6.11.

Theorem 6.12.

Proof.

Lemma 6.13.

Theorem 6.14.

Proof.

Remark 6.15.

Proposition 7.1.

Remark 7.2.

Theorem 7.3.

Proof.

Remark 7.4.

Remark 8.1.

Remark 8.2.

Arman Taghavi-Chabert Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland [email protected]

(Date: at \xxivtime)

We construct a one-parameter family of Lorentzian conformal structures on the canonical circle bundle of a partially integrable contact almost Cauchy–Riemann manifold. This builds on previous work by Leitner, who generalised Fefferman’s construction associated to a CR manifold to the non-involutive case. We provide characterisations of these conformal structures.

We introduce exact ‘perturbations’ of such Fefferman spaces by a semi-basic one-form, which can be suitably interpreted as a tuple of weighted tensors on the almost CR manifold. The resulting perturbed conformal space is an instance of a so-called nearly Robinson manifolds introduced recently by Fino, Leistner and the present author.

We investigate the existence of metrics in these conformal classes which satisfy appropriate sub-conditions of the Einstein equations. These metrics are defined only off cross-sections of Fefferman’s circle bundle, and are conveniently expressed in terms of almost Lorentzian densities, which include Gover’s almost Einstein scales as a special case. In particular, in dimensions greater than four, almost Einstein scales always arise from an CR–Einstein manifolds. We derive necessary and sufficient conditions for a perturbed Fefferman space to be conformally flat on the zero set of an almost Einstein scale.

An explicit example of a strictly almost CR–Einstein five-manifold fibred over a strictly almost Kähler–Einstein four-manifold due to Nurowski and Przanowski is constructed.

Conformal geometry, CR geometry, Lorentzian geometry, Fefferman spaces, Robinson geometry, optical geometry, congruences of null geodesics, Einstein metrics

Primary 53C18, 32V05, 53C50, 83C15; Secondary 53C15, 32V99, 53B30

This article is concerned with the interaction between Lorentzian conformal geometry and Cauchy–Riemann geometry, where the latter arises as the leaf space of a foliation by null geodesics on the former. This geometric setting can be found in two distinct and independent contexts. The first of these is due to Fefferman [Fef76a, Fef76] that associates a non-degenerate or contact CR structure of hypersurface type to a conformal structure of Lorentzian signature on a canonical circle bundle. There are various approaches to the construction: in the original version, Fefferman shows how a conformal structure can be constructed from the standard Kähler structure on the complex space $\mathbf{C}^{m+1}$ in which the CR hypersurface is embedded. Later formulations allow for the CR manifold $(\mathcal{M},H,J)$ to be abstract, that is, not necessarily embeddable. Of particular relevance in this article is the construction due to Lee [Lee86], who showed that Fefferman’s conformal structure lives on the total space of a circle bundle over $(\mathcal{M},H,J)$ , which is a certain quotient of the canonical bundle $\mathcal{C}$ over $\mathcal{M}$ . Čap and Gover [ČG08] later substituted $\mathcal{C}$ for a certain root thereof, which allows them to work within the framework of parabolic geometry and tractor calculus. In all of these versions, one associates to each contact form $\theta$ a so-called Fefferman metric

\displaystyle\widetilde{g}_{\theta}

\displaystyle=4\theta\odot\widetilde{\omega}^{W}_{\theta}+h\,,

(1.1)

where $\widetilde{\omega}^{W}_{\theta}$ is the induced Weyl connection one-form compatible with $\theta$ , and $h$ is the (degenerate) metric induced from the Levi form of $\theta$ . Conformally related contact forms yield conformal related Fefferman metrics. The conformal structure thus constructed is characterised by the existence of a null conformal Killing field, and the CR manifold is recovered as the leaf space of its space of integral curves.

Fefferman’s construction was later adapted by Leitner [Lei10] to the case where $(\mathcal{M},H,J)$ is a partially integrable non-degenerate almost CR manifolds — when $\mathcal{M}$ has dimension greater than three — and also to the case where the Weyl connection $\widetilde{\omega}^{W}_{\theta}$ in (1.1) is gauged by some horizontal one-form.

The other notion connecting Lorentzian geometry and almost Cauchy–Riemann geometry is known as almost Robinson geometry, an idea first introduced by Nurowski and Trautman in [NT02] in the involutive case, and developed further by Fino, Leistner and the present author in [FLTC23] in the more general case. An almost Robinson geometry is essential a Lorentzian analogue of an almost Hermitian geometry. This simply consists of a totally null complex $(m+1)$ -plane distribution $\widetilde{N}$ on a $(2m+2)$ -dimensional Lorentzian conformal manifold $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}})$ . This distribution intersects its complex conjugate in the complexification of a real null line distribution $\widetilde{K}$ . When $\widetilde{N}$ is preserved along any non-vanishing section $\widetilde{k}$ of $K$ , the integral curves of $\widetilde{k}$ are null geodesics, and $\widetilde{N}$ gives rise to an almost CR structure on the leaf space of the congruence generated by $\widetilde{k}$ . We then say that $\widetilde{N}$ is nearly Robinson. In dimension four, this setting is equivalent to a so-called optical geometry with non-shearing congruence of null geodesics. When the CR structure is contact, the congruence is said to be twisting. These congruences play a fundamental part in mathematical relativity, notably in the discovery of exact solutions to the vacuum Einstein’s field equations, such as the Kerr black hole [Ker63]. Aspects of Penrose’s twistor theory were highly influenced by these ideas [Pen67, Mas98]. At the same time, Robinson and Trautman [RT86] initiated a fruitful programme of research in that direction [Taf85, LN90, LNT91, HLN08].

In dimensions greater than four, there is a priori no connection between non-shearing congruences and nearly Robinson structures. The present author, however, recently [TC22] gave a mild curvature condition on the Weyl tensor for a twisting non-shearing congruence of null geodesics to give rise to a nearly Robinson structure. In the same reference, all local Einstein metrics on Lorentzian conformal manifolds of even dimension admitting a twisting non-shearing congruence of null geodesics were found to be contained in a three-parameter family of metrics, which include the Taub–NUT–(A)dS metric and Fefferman–Einstein metrics. In this case, the leaf space of the congruence is characterised by the existence of a CR–Einstein structure, which corroborates previous results in [Lei07] for CR geometries. Reference [ČG08] by Čap and Gover shed more lights on these aspects, especially in connection with the rôle played by so-called almost Einstein scales and their zero sets, which can be suitable identified with cross-sections of the Fefferman bundle.

Finally, there is an obvious analogy when it comes to the usefulness of Cauchy–Riemann methods in solving the Einstein equations on a Lorentzian manifold: if one wishes to find Einstein metrics on a Riemannian manifold, then more powerful resources become available once we assume that our manifold is Kähler — see for instance [Yau00] for a review of these ideas.

The overall aim of this paper is to recast some classes of nearly Robinson manifolds as Fefferman spaces of some kind. More precisely, we set ourselves the following goals:

(1)

To provide a formulation of CR–Einstein structures for non-involutive almost CR manifolds by means of the Webster calculus. These structures underlying a certain class of Einstein Lorentzian metrics to be considered here. Our investigation will also be illustrated by an example of a five-dimensional strictly almost CR manifold arising from a strictly almost Kähler–Einstein manifold.
(2)

To develop Leitner’s generalisation of Fefferman’s construction for almost CR manifolds by using the norm square of the Nijenhuis torsion tensor. This will lead to a one-parameter family of such conformal structures, and we will provide geometric characterisations for them, which generalises Sparling’s characterisation of Fefferman spaces for CR structures [Gra87, ČG08].
(3)

To introduce a new class of Lorentzian conformal structures arising from almost CR manifolds, as exact ‘perturbations’ of Fefferman conformal spaces: a ‘perturbed’ Fefferman metric takes the form $\widetilde{g}_{\theta,\widetilde{\xi}}=\widetilde{g}_{\theta}+4\theta\odot\widetilde{\xi}$ , where $g_{\theta}$ is a Fefferman metric for some pseudo-Hermitian form $\theta$ , and $\widetilde{\xi}$ is a semi-basic one-form.
(4)

To seek almost Einstein scales for such perturbed Fefferman spaces in dimensions greater than four, and investigate the geometric properties of their zero sets.

Let us re-emphasise that the focus of this article will be on conformal geometries of dimensions greater than four, although some results may still apply in dimension four. It turns out that the four-dimensional story is somewhat richer and more technical, and for this reason, it is treated separately in [TC23].

The plan of the article is as follows. We start in Section 2 with a review of almost CR geometry, focussing in particular on so-called CR–Einstein structures. In Theorem 2.4, these pseudo-Hermitian analogue of the Einstein equations are shown to be equivalent to solutions to a system of invariant differential equations on a CR scale. In Theorem 2.8, a variation of these equations, now involving a complex density, is proposed, and a solution defines a CR–Einstein scale, which turns out to be a constant multiple of the norm squared of the Nijenhuis tensor — Proposition 2.10.

Section 3 contains a review of conformal geometry and so-called optical and almost Robinson geometries which are at the heart of non-shearing congruences of null geodesics. In Definitions 3.3 and 3.7, we introduce formal definitions of certain metrics with prescribed Ricci curvature, which we term weakly half-Einstein, half-Einstein and pure radiation metrics, and their almost Lorentzian scale analogues as generalisations of almost Einstein scales — Proposition 3.10.

In Section 4, we construct a one-dimensional family of Lorentzian conformal structures from a strictly almost CR manifold, by modifying the original Fefferman construction — see Definitions 4.1 and 4.2. We then state the characterisation of these conformal structures that arise from such Fefferman spaces in Theorem 4.6.

Section 5 begins with the formal Definition 5.1 of perturbations of Fefferman spaces. We then provide a technical Lemma 5.7, which allows to formalise the relation between perturbed Fefferman spaces and geometries endowed with twisting non-shearing congruences of null geodesics. Propositions 5.11 provides sufficient conditions for an optical geometry to be locally conformally isometric to a perturbed Fefferman space. The section ends with Conjecture 5.14, which puts forward another characterisation of certain classes of perturbed Fefferman spaces.

The focus of Section 6 is to examine the consequences of the existence of almost Lorentzian scales on optical geometries with twisting non-shearing congruences of null geodesics. Among others, Proposition 6.1 provide geometric descriptions of the zero set of some classes of such scales, while Propositions 6.9 and 6.10 provide characterisations of almost (weakly) half-Einstein scales in terms of their underlying CR geometries. In Corollary 6.11 we give sufficient conditions for an optical geometry to be locally conformally isometric to a perturbed Fefferman space. The CR data of perturbed Fefferman spaces admitting almost (half) Einstein scales is described in Theorems 6.12 and 6.14.

Section 7 looks into the properties the zero set of an almost half-Einstein scale on a perturbed Fefferman space, more particularly, Proposition 7.1 in relation to its causal property, and Theorem 7.3 in relation to conformal flatness.

We end the article with Section 8 by commenting briefly on conformal symmetries and their interpretation in terms of CR invariant differential equations.

The article ends with two appendices: In Appendix A we give an example of an strictly almost CR–Einstein manifold in dimension five, and Appendix B contains the lengthy proof of Theorem 4.6.

Acknowledgments:The research leading to these results has received funding from the Norwegian Financial Mechanism 2014-2021 UMO-2020/37/K/ST1/02788.

We first recall some background on almost CR geometry following [Tan75, Web78, GG05, ČG08, ČG10, Mat16, CG20, Mat22] before introducing more novel ideas in Sections 2.4 and 2.5.

An almost Cauchy–Riemann (CR) structure on a smooth manifold $\mathcal{M}$ of dimension $2m+1$ consists of a pair $(H,J)$ where $H$ is a rank- $2m$ distribution and ${J}$ a bundle complex structure on ${H}$ , i.e. ${J}\circ{J}=-{\mathrm{Id}}$ , where ${\mathrm{Id}}$ is the identity map on ${H}$ . The complexification ${}^{\mathbf{C}}{H}$ of ${H}$ thus splits as ${}^{\mathbf{C}}{H}={H}^{(1,0)}\oplus{H}^{(0,1)}$ , where ${H}^{(1,0)}$ and ${H}^{(0,1)}$ are the $\mathrm{i}$ -eigenbundle and $-\mathrm{i}$ -eigenbundle of ${J}$ respectively, each having complex rank $m$ . Throughout this article, the following additional assumptions will be made:

•

$(H,J)$ is partially integrable, that is, $[H^{(1,0)},H^{(1,0)}]\subset{}^{\mathbf{C}}H$ ;
•

$H$ is contact, that is, $T\mathcal{M}=H+[H,H]$ .

We shall refer to the triple $(\mathcal{M},H,J)$ thus defined as an almost CR manifold. A choice of contact form $\theta\in\Gamma(\mathrm{Ann}(H))$ will be referred to as pseudo-hermitian structure for $(H,J)$ . The restriction of $\mathrm{d}\theta$ to $H$ is non-degenerate, and is the imaginary part of a hermitian bilinear form called the Levi form associated to $\theta$ . Its signature $(p,q)$ , where $p+q=2m$ , is an invariant of $(H,J)$ .

We shall also assume that the canonical bundle ${\mathcal{C}}:=\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}({H}^{(0,1)})$ of $(H,J)$ admits an $-(m+2)$ -nd root, which we shall denote ${\mathcal{E}}(1,0)$ . More generally, we define density bundles ${\mathcal{E}}(w,w^{\prime}):={\mathcal{E}}(1,0)^{w}\otimes\overline{{\mathcal{E}}(1,0)}{}^{w^{\prime}}$ for any $w,w^{\prime}\in\mathbf{C}$ such that $w-w^{\prime}\in\mathbf{Z}$ . We note that $\overline{{\mathcal{E}}(w,w^{\prime})}={\mathcal{E}}(w^{\prime},w)$ . In particular, $\mathcal{C}=\mathcal{E}(-m-2,0)$ .

There are non-vanishing canonical sections ${\bm{\theta}}$ of $T^{*}{\mathcal{M}}\otimes{\mathcal{E}}(1,1)$ and ${\bm{h}}$ of $({H}^{(1,0)})^{*}\otimes({H}^{(0,1)})^{*}\otimes{\mathcal{E}}(1,1)$ with the property that for each real $0<s\in\Gamma({\mathcal{E}}(1,1))$ , ${\theta}=s^{-1}{\bm{\theta}}$ is a pseudo-hermitian structure with Levi form ${h}=s{\bm{h}}$ . The weighted form $\bm{h}$ is called the Levi form of $(H,J)$ . We shall refer to $s$ as a CR scale. We shall also identify ${H}^{(1,0)}$ with $({H}^{(0,1)})^{*}(1,1)$ and ${H}^{(0,1)}$ with $({H}^{(1,0)})^{*}(1,1)$ by means of $\bm{h}$ and its inverse.

A density $\sigma$ of weight $(1,0)$ determines a unique CR scale $s=\sigma\overline{\sigma}$ , which in turn gives rise to a pseudo-hermitian structure ${\theta}=s^{-1}{\bm{\theta}}$ . Conversely, at any point, a CR scale determines a circle of densities of weight $(1,0)$ . If $\sigma$ is such a density, then the section $\zeta:=\sigma^{-(m+2)}$ of the canonical bundle satisfies

\displaystyle\theta\wedge(\mathrm{d}\theta)^{m}

\displaystyle=\mathrm{i}^{m^{2}}m!(-1)^{q}\theta\wedge(\ell\makebox[7.0pt]{\rule{6.0pt}{0.3pt}\rule{0.3pt}{5.0pt}}\,\zeta)\wedge(\ell\makebox[7.0pt]{\rule{6.0pt}{0.3pt}\rule{0.3pt}{5.0pt}}\,\overline{\zeta})\,.

(2.1)

and is said to be volume-normalised.

We introduce plain and barred minuscule Greek abstract indices in the following way:

	$\displaystyle{\mathcal{E}}^{\alpha}$	$\displaystyle:={H}^{(1,0)}\,,$	$\displaystyle{\mathcal{E}}^{\bar{\alpha}}$	$\displaystyle:={H}^{(0,1)}\,,$	$\displaystyle{\mathcal{E}}_{0}$	$\displaystyle:=\mathrm{Ann}(H)\,,$
	$\displaystyle{\mathcal{E}}_{\alpha}$	$\displaystyle:=({H}^{(1,0)})^{*}\,,$	$\displaystyle{\mathcal{E}}_{\bar{\alpha}}$	$\displaystyle:=({H}^{(0,1)})^{*}\,,$	$\displaystyle{\mathcal{E}}^{0}$	$\displaystyle:=\left(\mathrm{Ann}(H)\right)^{*}\,.$

In effect, indices can be raised and lowered using ${\bm{h}}_{\alpha\bar{\beta}}$ , e.g. ${v}_{\alpha}={\bm{h}}_{\alpha\bar{\beta}}{v}^{\bar{\beta}}$ . Clearly, complex conjugation on ${}^{\mathbf{C}}{H}$ changes the index type, so we shall write ${v}^{\bar{\alpha}}$ for $\overline{{v}^{\alpha}}$ , and so on.

Symmetrisation will be denoted by round brackets, and skew-symmetrisation by square brackets, e.g. $\lambda_{(\alpha\beta)}=\frac{1}{2}\left(\lambda_{\alpha\beta}+\lambda_{\beta\alpha}\right)$ and $\lambda_{[\alpha\beta]}=\frac{1}{2}\left(\lambda_{\alpha\beta}-\lambda_{\beta\alpha}\right)$ . The tracefree part of tensors of mixed types with respect to the Levi form will be denoted by a ring, e.g. $(\lambda_{\alpha\bar{\beta}})_{\circ}=\lambda_{\alpha\bar{\beta}}-\frac{1}{m}\bm{h}_{\alpha\bar{\beta}}\lambda_{\gamma\bar{\delta}}\bm{h}^{\gamma\bar{\delta}}$ .

For each choice of pseudo-hermitian structure ${\theta}$ , there is a unique vector field ${\ell}$ , known as the Reeb vector field, which satisfies ${\theta}({\ell})=1$ and $\mathrm{d}{\theta}({\ell},\cdot)=0$ . In particular, it induces a splitting

\displaystyle{}^{\mathbf{C}}T{\mathcal{M}}={}^{\mathbf{C}}\langle\ell\rangle\oplus{H}^{(1,0)}\oplus{H}^{(0,1)}\,.

Any choice frame $(e_{\alpha})_{\alpha=1,\ldots,m}$ of $H^{(1,0)}$ can thus be completed to an adapted frame $\left(\ell,{e}_{\alpha},\overline{{e}}_{\bar{\beta}}\right)$ , $\alpha,\bar{\beta}=1,\ldots,m$ for ${}^{\mathbf{C}}T\mathcal{M}$ . Similarly, we call its dual $\left({\theta},{\theta}^{\alpha},\overline{{\theta}}{}^{\bar{\alpha}}\right)$ an adapted coframe for ${}^{\mathbf{C}}T^{*}\mathcal{M}$ .

The structure equations obtained from this adapted coframe can then be written as


$\displaystyle\mathrm{d}{\theta}$	$\displaystyle=\mathrm{i}{h}_{\alpha\bar{\beta}}{\theta}^{\alpha}\wedge\overline{{\theta}}{}^{\bar{\beta}}\,,$	(2.2a)
$\displaystyle\mathrm{d}{\theta}^{\alpha}$	$\displaystyle={\theta}^{\beta}\wedge{\Gamma}_{\beta}{}^{\alpha}+{\mathsf{A}}^{\alpha}{}_{\bar{\beta}}{\theta}\wedge\overline{{\theta}}{}^{\bar{\beta}}-\frac{1}{2}{\mathsf{N}}_{\bar{\beta}\bar{\gamma}}{}^{\alpha}\overline{{\theta}}{}^{\bar{\beta}}\wedge\overline{{\theta}}{}^{\bar{\gamma}}\,,$	(2.2b)
$\displaystyle\mathrm{d}\overline{{\theta}}{}^{\bar{\alpha}}$	$\displaystyle=\overline{{\theta}}{}^{\bar{\beta}}\wedge{\Gamma}_{\bar{\beta}}{}^{\bar{\alpha}}+{\mathsf{A}}^{\bar{\alpha}}{}_{\beta}{\theta}\wedge{\theta}^{\beta}-\frac{1}{2}{\mathsf{N}}_{\beta\gamma}{}^{\bar{\alpha}}\theta^{\beta}\wedge\theta^{\gamma}\,,$	(2.2c)

where

•

${h}_{\alpha\bar{\beta}}$ is the Levi form of ${\theta}$ , viewed as Hermitian matrix — abstractly, this is $h_{\alpha\bar{\beta}}=s^{-1}\bm{h}_{\alpha\bar{\beta}}$ where $s$ is the CR scale of $\theta$ ;

•

${\Gamma}_{\beta}{}^{\alpha}$ is the connection $1$ -form of the Webster connection ${\nabla}$ on $T{\mathcal{M}}$ that preserves $\theta$ and $\mathrm{d}\theta$ , i.e.

\displaystyle{\nabla}{\theta}

\displaystyle=0\,,

\displaystyle{\nabla}{\theta}^{\alpha}

\displaystyle=-{\Gamma}_{\beta}{}^{\alpha}\otimes{\theta}^{\beta}\,,

\displaystyle{\nabla}\overline{{\theta}}{}^{\bar{\alpha}}

\displaystyle=-\Gamma_{\bar{\beta}}{}^{\bar{\alpha}}\otimes\overline{\theta}{}^{\bar{\beta}}\,,

with $\mathrm{d}{h}_{\alpha\bar{\beta}}-\Gamma_{\alpha\bar{\beta}}-\Gamma_{\bar{\beta}\alpha}=0$ ;

•

${\mathsf{A}}_{\alpha\beta}$ is the Webster torsion tensor of $\nabla$ , which satisfies ${\mathsf{A}}_{\alpha\beta}={\mathsf{A}}_{(\alpha\beta)}$ ;
•

and ${\mathsf{N}}_{\alpha\beta\gamma}$ is the Nijenhuis torsion tensor of $\nabla$ , which satisfies ${\mathsf{N}}_{\alpha\beta\gamma}={\mathsf{N}}_{[\alpha\beta]\gamma}$ , ${\mathsf{N}}_{[\alpha\beta\gamma]}=0$ .

The Webster connection, also known as Webster–Tanaka or Webster–Stanton connection, is uniquely determined by its compatibility with the contact form and its Levi form, and the prescription of symmetry on its torsion.

Clearly, the Nijenhuis tensor ${\mathsf{N}}_{\alpha\beta\gamma}$ vanishes identically if and only if $H^{(1,0)}$ is involutive. This is a CR invariant condition. For convenience, we set

\displaystyle\|\mathsf{N}\|^{2}

\displaystyle:=\mathsf{N}_{\alpha\beta\gamma}\mathsf{N}^{\alpha\beta\gamma}\,.

This is a density of weight $(-1,-1)$ . If the Levi form has definite signature, i.e. $pq=0$ , then its vanishing is equivalent to the vanishing of $\mathsf{N}_{\alpha\beta\gamma}$ , and thus to the involutivity of $(H,J)$ .

On the other hand, the Reeb vector field $\ell$ of $(H,J,\theta)$ is an infinitesimal symmetry of $({H},{J})$ if and only if ${\mathsf{A}}_{\alpha\beta}$ vanishes identically.

The curvature tensors of $\nabla$ are given by, for any section ${V}^{\alpha}$ of ${H}^{(1,0)}$ ,

	$\displaystyle({\nabla}_{\alpha}{\nabla}_{\bar{\beta}}-{\nabla}_{\bar{\beta}}{\nabla}_{\alpha}){V}^{\gamma}+\mathrm{i}\bm{{h}}_{\alpha\bar{\beta}}{V}^{\gamma}$	$\displaystyle=:{\mathsf{R}}_{\alpha\bar{\beta}\delta}{}^{\gamma}{V}^{\delta}\,,$
	$\displaystyle({\nabla}_{\alpha}{\nabla}_{0}-{\nabla}_{0}{\nabla}_{\alpha}){V}^{\gamma}-{\mathsf{A}}_{\alpha}{}^{\bar{\beta}}\nabla_{\bar{\beta}}{V}^{\gamma}$	$\displaystyle=:{\mathsf{R}}_{\alpha 0\delta}{}^{\gamma}{V}^{\delta}\,,$
	$\displaystyle({\nabla}_{\alpha}{\nabla}_{\beta}-{\nabla}_{\beta}{\nabla}_{\alpha}){V}^{\gamma}-{\mathsf{N}}_{\alpha\beta}{}^{\bar{\delta}}\nabla_{\bar{\delta}}{V}^{\gamma}$	$\displaystyle=:{\mathsf{R}}_{\alpha\beta\delta}{}^{\gamma}{V}^{\delta}\,,$

together with their complex conjugates. The Bianchi identities allows us to express ${\mathsf{R}}_{\alpha 0\delta}{}^{\gamma}$ and ${\mathsf{R}}_{\alpha\beta\delta}{}^{\gamma}$ in terms of the torsion and its covariant derivatives as given in [TC22]. The more important piece ${\mathsf{R}}_{\alpha\bar{\beta}\gamma}{}^{\delta}$ can be used to define, for $m>1$ , the Chern–Moser tensor

\displaystyle{\mathsf{S}}_{\alpha}{}^{\gamma}{}_{\beta}{}^{\delta}

\displaystyle:=\left({\mathsf{R}}_{(\alpha}{}^{(\gamma}{}_{\beta)}{}^{\delta)}\right)_{\circ}\,,

and for $m\geq 1$ , the Webster–Ricci tensor ${\mathsf{Ric}}{}_{\gamma}{}^{\delta}:={\bm{h}}{}^{\alpha\bar{\beta}}{\mathsf{R}}{}_{\alpha\bar{\beta}\gamma}{}^{\delta}$ and the Webster–Ricci scalar ${\mathsf{Sc}}:={\mathsf{Ric}}{}_{\gamma}{}^{\gamma}$ . We conveniently introduce the Webster–Schouten tensor and the Webster–Schouten scalar

\displaystyle{\mathsf{P}}_{\alpha\bar{\beta}}

\displaystyle:=\frac{1}{m+2}\left({\mathsf{Ric}}_{\alpha\bar{\beta}}-\frac{1}{2m+2}{\mathsf{Sc}}\,{\bm{h}}_{\alpha\bar{\beta}}\right)\,,

\displaystyle{\mathsf{P}}:={\mathsf{P}}_{\alpha\bar{\beta}}{\bm{h}}^{\alpha\bar{\beta}}\,,

respectively. It then follows that $\mathsf{P}=\frac{1}{2(m+1)}\mathsf{Sc}$ .

The Chern–Moser tensor is a CR invariant. In dimension greater than three, the vanishing of both ${\mathsf{N}}_{\alpha\beta\gamma}$ and ${\mathsf{S}}_{\alpha\bar{\gamma}\beta\bar{\delta}}$ is equivalent to the almost CR structure being locally CR flat — if $pq=0$ , this means that ${\mathcal{M}}$ is locally equivalent to the CR $(2m+1)$ -sphere.

The Webster connection extends to a connection on the density bundles $\mathcal{E}(w,w^{\prime})$ . If $\sigma\in\Gamma(\mathcal{E}(w,0))$ is such that $\sigma^{-\frac{m+2}{w}}=\theta\wedge\theta^{1}\wedge\ldots\wedge\theta^{m}$ for some adapted coframe $(\theta,\theta^{\alpha},\overline{\theta}{}^{\bar{\alpha}})$ . Then

\displaystyle\nabla\sigma

\displaystyle=\frac{w}{m+2}\Gamma_{\alpha}{}^{\alpha}\sigma\,,

(2.3)

where $\Gamma_{\alpha}{}^{\beta}$ is the connection one-form of $\nabla$ corresponding to $\theta$ . By complex conjugation, one obtains a similar formula for densities of weight $(0,w)$ .

Note that $\nabla$ preserves the weighted contact form $\bm{\theta}$ , i.e. $\nabla\bm{\theta}=0$ . Thus if $\theta=s^{-1}\bm{\theta}$ for some CR scale $s$ , $\nabla$ also preserves $s$ . This implies that for any $\sigma\in\Gamma(\mathcal{E}(1,0))$ such that $s=\sigma\overline{\sigma}$ , we must have $\sigma^{-1}\nabla\sigma=-\overline{\sigma}^{-1}\nabla\overline{\sigma}$ .

For any smooth density $f\in\Gamma(\mathcal{E}(w,w^{\prime}))$ , the commutation relations are given by


$\displaystyle({\nabla}_{\alpha}{\nabla}_{\bar{\beta}}-{\nabla}_{\bar{\beta}}{\nabla}_{\alpha}){f}$	$\displaystyle=\frac{w-w^{\prime}}{m+2}\left(\mathsf{Ric}_{\alpha\bar{\beta}}-\mathsf{N}_{\gamma\delta\alpha}\mathsf{N}^{\gamma\delta}{}_{\bar{\beta}}+\mathsf{N}_{\alpha\gamma\delta}{\mathsf{N}}_{\bar{\beta}}{}^{\gamma\delta}\right)f-\mathrm{i}\bm{{h}}_{\alpha\bar{\beta}}{\nabla}_{0}{f}\,,$	(2.4a)
$\displaystyle({\nabla}_{\alpha}{\nabla}_{0}-{\nabla}_{0}{\nabla}_{\alpha}){f}$	$\displaystyle=\frac{w-w^{\prime}}{m+2}\left({\nabla}^{\beta}{\mathsf{A}}_{\alpha\beta}-{\mathsf{A}}^{\beta\gamma}{\mathsf{N}}_{\alpha\beta\gamma}\right)f+{\mathsf{A}}_{\alpha}{}^{\bar{\beta}}{\nabla}_{\bar{\beta}}{f}\,,$	(2.4b)
$\displaystyle({\nabla}_{\alpha}{\nabla}_{\beta}-{\nabla}_{\beta}{\nabla}_{\alpha}){f}$	$\displaystyle=\frac{w-w^{\prime}}{m+2}{\nabla}^{\gamma}{\mathsf{N}}_{\alpha\beta\gamma}f+{\mathsf{N}}_{\alpha\beta}{}^{\bar{\gamma}}{\nabla}_{\bar{\gamma}}{f}\,.$	(2.4c)

This follows from the trace of the computation of the curvature $2$ -form $\Omega_{\gamma}{}^{\gamma}=\mathrm{d}\Gamma_{\gamma}{}^{\gamma}$ , which we find to be

\Omega_{\gamma}{}^{\gamma}=\left(\mathsf{Ric}_{\alpha\bar{\beta}}-\mathsf{N}_{\gamma\delta\alpha}\mathsf{N}^{\gamma\delta}{}_{\bar{\beta}}+\mathsf{N}_{\alpha\gamma\delta}{\mathsf{N}}_{\bar{\beta}}{}^{\gamma\delta}\right){\theta}^{\alpha}\wedge\overline{{\theta}}{}^{\bar{\beta}}\\ +\frac{1}{2}{\nabla}^{\alpha}{\mathsf{N}}_{\gamma\delta\alpha}{\theta}^{\gamma}\wedge{\theta}^{\delta}-\frac{1}{2}{\nabla}^{\bar{\alpha}}{\mathsf{N}}_{\bar{\gamma}\bar{\delta}\bar{\alpha}}\overline{{\theta}}{}^{\bar{\gamma}}\wedge\overline{{\theta}}{}^{\bar{\delta}}\\ +\mathrm{i}\left((m+2)T_{\gamma}-\nabla_{\gamma}\mathsf{P}\right){\theta}^{\gamma}\wedge{\theta}^{0}-\mathrm{i}\left((m+2)T_{\bar{\gamma}}-\nabla_{\bar{\gamma}}\mathsf{P}\right)\overline{{\theta}}{}^{\bar{\gamma}}\wedge{\theta}^{0}\,.

(2.5)

where

\displaystyle{T}_{\alpha}

\displaystyle=\frac{1}{m+2}\left({\nabla}_{\alpha}{\mathsf{P}}-\mathrm{i}{\nabla}^{\gamma}{\mathsf{A}}_{\gamma\alpha}+\mathrm{i}{\mathsf{A}}^{\beta\gamma}{\mathsf{N}}_{\alpha\beta\gamma}\right)\,.

(2.6)

Under a change of contact forms, the Webster connection is subject to transformation laws. These can be found in e.g. [Mat22], and do not differ from the involutive case, see e.g. [GG05]. It will suffice to state the transformation rules of the Webster torsion, Webster–Schouten tensor and Webster–Schouten scalar as

$\displaystyle\widehat{{\mathsf{A}}}_{\alpha\beta}$	$\displaystyle={\mathsf{A}}_{\alpha\beta}+\mathrm{i}{\nabla}_{(\alpha}{\Upsilon}_{\beta)}-\mathrm{i}{\Upsilon}_{\alpha}{\Upsilon}_{\beta}+\mathrm{i}\Upsilon^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}\,,$	(2.7)
$\displaystyle\widehat{\mathsf{P}}_{\alpha\bar{\beta}}$	$\displaystyle=\mathsf{P}_{\alpha\bar{\beta}}-\frac{1}{2}\left(\nabla_{\alpha}\Upsilon_{\bar{\beta}}+\nabla_{\bar{\beta}}\Upsilon_{\alpha}\right)-\frac{1}{2}\Upsilon^{\gamma}\Upsilon_{\gamma}h_{\alpha\bar{\beta}}\,,$	(2.8)
$\displaystyle\widehat{{\mathsf{P}}}$	$\displaystyle={\mathsf{P}}-\frac{1}{2}\left({\nabla}^{\alpha}{\Upsilon}_{\alpha}+{\nabla}_{\alpha}{\Upsilon}^{\alpha}\right)-\frac{m}{2}{\Upsilon}^{\gamma}{\Upsilon}_{\gamma}\,,$	(2.9)

respectively. For future use, we also compute

\displaystyle\widehat{\nabla}_{\bar{\delta}}\mathsf{N}_{\gamma\alpha\beta}

\displaystyle=\nabla_{\bar{\delta}}\mathsf{N}_{\gamma\alpha\beta}+\Upsilon_{\bar{\delta}}\mathsf{N}_{\gamma\alpha\beta}+h_{\gamma\bar{\delta}}\mathsf{N}_{\varepsilon\alpha\beta}\Upsilon^{\varepsilon}+h_{\alpha\bar{\delta}}\mathsf{N}_{\gamma\varepsilon\beta}\Upsilon^{\varepsilon}+h_{\beta\bar{\delta}}\mathsf{N}_{\gamma\alpha\varepsilon}\Upsilon^{\varepsilon}\,,

and hence

\displaystyle\widehat{\nabla}^{\gamma}\mathsf{N}_{\gamma[\alpha\beta]}

\displaystyle=\nabla^{\gamma}\mathsf{N}_{\gamma[\alpha\beta]}+(m+2)\Upsilon^{\gamma}\mathsf{N}_{\gamma[\alpha\beta]}\,,

\displaystyle\widehat{\nabla}^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}

\displaystyle=\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}+m\Upsilon^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}\,.

(2.10)

In what follows, $\mathcal{E}_{\bullet}$ will denote any multiple tensor products of $\mathcal{E}_{\alpha}$ and $\mathcal{E}_{\bar{\alpha}}$ . Choose a Webster connection $\nabla$ . Then, for any choice of one-form $\xi$ on $\mathcal{M}$ , we define the connection

\displaystyle\accentset{\xi}{\nabla}

\displaystyle:\Gamma(\mathcal{E}_{\bullet}(w,w^{\prime}))\longrightarrow\Gamma(T^{*}\mathcal{M}\otimes\mathcal{E}_{\bullet}(w,w^{\prime}))\quad:\bm{f}\mapsto\accentset{\xi}{\nabla}\bm{f}:=\nabla\bm{f}-\mathrm{i}(w-w^{\prime})\xi\otimes\bm{f}\,.

(2.11)

Clearly, $\accentset{\xi}{\nabla}$ coincides with $\nabla$ on any section of $\mathcal{E}_{\bullet}(w,w)$ . We shall often use the short-hand $\accentset{\xi}{\nabla}=\nabla+\xi$ to mean (2.11).

We shall call the connection defined by (2.11) the Webster connection gauged by $\xi$ . If $\xi=\mathrm{i}\nabla\log z$ for some complex-valued function $z$ , we refer to the gauge $\xi$ as being exact.

Let $\xi$ be a one-form on $(\mathcal{M},H,J)$ so that $\xi=\xi_{\alpha}\theta^{\alpha}+\xi_{\bar{\alpha}}\overline{\theta}^{\bar{\alpha}}+\xi_{0}\theta$ for some adapted coframe $(\theta,\theta^{\alpha},\overline{\theta}{}^{\bar{\alpha}})$ . A sufficient and necessary condition for $\xi$ to be closed is that

\displaystyle\nabla_{[\alpha}\xi_{\beta]}-\frac{1}{2}\mathsf{N}_{\alpha\beta}{}^{\bar{\gamma}}\xi_{\bar{\gamma}}

\displaystyle=0\,,

\displaystyle\nabla_{\alpha}\xi_{\bar{\beta}}-\nabla_{\bar{\beta}}\xi_{\alpha}+\mathrm{i}\xi_{0}{h}_{\alpha\bar{\beta}}

\displaystyle=0\,,

\displaystyle\nabla_{0}\xi_{\beta}-\nabla_{\beta}\xi_{0}+\xi_{\bar{\alpha}}{\mathsf{A}}^{\bar{\alpha}}{}_{\beta}

\displaystyle=0\,.

It suffices to compute the exterior derivative of $\xi$ :

\displaystyle F

\displaystyle:=\mathrm{d}\xi=F_{\alpha\beta}\theta^{\alpha}\wedge\theta^{\beta}+F_{\bar{\alpha}\bar{\beta}}\overline{\theta}{}^{\bar{\alpha}}\wedge\overline{\theta}{}^{\bar{\beta}}+2F_{\alpha\bar{\beta}}\theta^{\alpha}\wedge\overline{\theta}{}^{\bar{\beta}}+2F_{0\beta}\theta\wedge\theta^{\beta}+2F_{0\bar{\beta}}\theta\wedge\overline{\theta}{}^{\bar{\beta}}\,,

where, using the structure equations (2.2),

	$\displaystyle F_{\alpha\beta}$	$\displaystyle=\nabla_{[\alpha}\xi_{\beta]}-\frac{1}{2}\mathsf{N}_{\alpha\beta}{}^{\bar{\gamma}}\xi_{\bar{\gamma}}\,,$	$\displaystyle F_{\alpha\bar{\beta}}$	$\displaystyle=\frac{1}{2}\left(\nabla_{\alpha}\xi_{\bar{\beta}}-\nabla_{\bar{\beta}}\xi_{\alpha}+\mathrm{i}\xi_{0}{h}_{\alpha\bar{\beta}}\right)\,,$
	$\displaystyle F_{0\beta}$	$\displaystyle=\frac{1}{2}\left(\nabla_{0}\xi_{\beta}-\nabla_{\beta}\xi_{0}+\xi_{\bar{\alpha}}{\mathsf{A}}^{\bar{\alpha}}{}_{\beta}\right)\,.$

The result follows immediately. ∎

A CR–Einstein structure¹¹1Such a structure was introduced by the author in [TC22] under the name almost CR–Einstein structure. However, the use of the word ‘almost’ clashes with that in the notion of almost CR–Einstein structure given in [ČG08]. For this reason, it was deemed appropriate to drop the adverb ‘almost’ to avoid confusion. That $(H,J)$ is not necessarily involutive should be clear from the context. on an almost CR manifold $({\mathcal{M}},{H},{J})$ is a pseudo-hermitian structure, whose Webster torsion tensor ${\mathsf{A}}_{\alpha\beta}$ , Webster–Schouten tensor ${\mathsf{P}}_{\alpha\bar{\beta}}$ and Nijenhuis tensor ${\mathsf{N}}_{\alpha\beta\gamma}$ satisfy


	$\displaystyle\mathsf{A}_{\alpha\beta}=0\,,$		(2.12a)
	$\displaystyle\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}=0\,,$		(2.12b)
	$\displaystyle\left(\mathsf{P}_{\alpha\bar{\beta}}-\frac{1}{m+2}\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}\right)_{\circ}=0\,.$		(2.12c)

In the same reference, it is proved that condition (2.12c) can be replaced by

\displaystyle\mathsf{Ric}_{\alpha\bar{\beta}}-\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}

\displaystyle=\Lambda h_{\alpha\bar{\beta}}\,,

for some constant

\Lambda

In particular,

\displaystyle\Lambda

\displaystyle=\frac{1}{m}\left(\mathsf{Sc}-\|\mathsf{N}\|^{2}\right)\,.

(2.13)

When $(H,J)$ is involutive, equations (2.12) reduce to

\displaystyle\mathsf{A}_{\alpha\beta}=0\,,

\displaystyle\left(\mathsf{P}_{\alpha\bar{\beta}}\right)_{\circ}=0\,,

(2.14)

which were first investigated in [Lei07], where the corresponding pseudo-Hermitian structure is then referred to as a transversally symmetric pseudo-Einstein structure. These were later revisited in tractor language in [ČG08] under the name CR–Einstein structure. There, building on a result by [Lee88], the authors showed that the CR–Einstein equations (2.14) are equivalent to the existence of a density $\sigma$ of weight $(1,0)$ satisfying the system of CR-invariant differential equations


$\displaystyle\nabla_{\bar{\alpha}}\sigma$	$\displaystyle=0\,,$	(2.15a)
$\displaystyle\nabla_{\alpha}\nabla_{\beta}\sigma+\mathrm{i}\mathsf{A}_{\alpha\beta}$	$\displaystyle=0\,.$	(2.15b)

If $\sigma$ is allowed to have a non-empty zero set, the resulting structure is called almost CR–Einstein. The prolongation of the system (2.15) leads to the construction of the so-called CR tractor bundle, that is, a complex rank- $(m+2)$ vector bundle, equipped with the so-called normal tractor connection. In fact, any solution to (2.14) can be equivalently encoded as a parallel section of the CR tractor bundle — see [ČG08] for details.

In [TC23], where $(\mathcal{M},H,J)$ is assumed to be involutive, densities satisfying condition (2.15a) are referred to CR densities in analogy to CR functions. These are shown to be equivalent to the existence of a closed section of the canonical bundle, which is always guaranteed for realisable CR manifolds.

Such an interpretation clearly does not hold in the non-involutive case however. Certainly, strictly almost CR manifolds cannot admit closed section of the canonical bundle as the structure equations will immediately reveal. However, CR densities in the sense of (2.15a) may still exist on strictly almost CR manifolds as the example in Appendix A shows.

We may nevertheless re-express the almost CR–Einstein conditions (2.12) in terms of a CR scale:

A CR–Einstein structure on an almost CR manifold $(\mathcal{M},H,J)$ determines and is determined by a unique CR scale $s$ that satisfies the system of CR invariant differential equations:


	$\displaystyle\nabla_{(\alpha}\nabla_{\beta)}s+\mathrm{i}\mathsf{A}_{\alpha\beta}s+\mathsf{N}_{\gamma(\alpha\beta)}\nabla^{\gamma}s=0\,,$		(2.16a)
	$\displaystyle\nabla^{\gamma}s\mathsf{N}_{\gamma(\alpha\beta)}+\frac{1}{m}\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}s=0\,,$		(2.16b)
	$\displaystyle\left(s\nabla_{\bar{\beta}}\nabla_{\alpha}s-\nabla_{\alpha}s\nabla_{\bar{\beta}}s+\mathsf{P}_{\alpha\bar{\beta}}s^{2}-\frac{1}{m+2}\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}s^{2}\right)_{\circ}=0\,.$		(2.16c)

Choosing $\nabla$ such that $\nabla s=0$ immediately gives us the implication (2.16) $\Rightarrow$ (2.12). For the converse, it is enough to use the transformation laws for $\mathsf{A}_{\alpha\beta}$ , $\nabla^{\gamma}\mathsf{N}_{\gamma\alpha\beta}$ and $\mathsf{P}_{\alpha\bar{\beta}}$ given by (2.7), (2.10) and (2.8) with $\Upsilon=-s^{-1}\nabla s$ . The form of (2.16c) can then be obtained by applying the commutation relation (2.4a). ∎

Let $(\mathcal{M},H,J)$ be an almost CR manifold. We shall call any real density $s$ of weight $(1,1)$ that is a solution of the system (2.16) an almost CR–Einstein scale. If $s$ is a CR scale (and so, nowhere vanishing), it will be referred to simply as a CR–Einstein scale.

In this article, we shall however only be concerned with solutions that are CR scales.

Equation (2.16a) already featured in [Mat22] in the context of deformation of almost CR structures. Using (2.4c), it can be recast as

\displaystyle\nabla_{\alpha}\nabla_{\beta}s+\mathrm{i}\mathsf{A}_{\alpha\beta}s+\mathsf{N}_{\gamma\alpha\beta}\nabla^{\gamma}s

\displaystyle=0\,.

We now turn to the geometric interpretation of each of the conditions given in the previous theorem. That (2.16a) characterises the existence of a CR symmetry is clear, since it defines a pseudo-Hermitian structure for which (2.12a) holds, i.e. the Reeb vector field is an infinitesimal CR symmetry. In addition, applying Corollary 4.1 of [TC22] yields the following.

Let $(\mathcal{M},H,J)$ be an almost CR manifold, and let $s\in\Gamma(\mathcal{E}(1,1))$ be a CR scale. Then $s$ satisfies (2.16a) if and only if the Reeb vector field of the contact form $\theta=s^{-1}\bm{\theta}$ is an infinitesimal CR symmetry. This being the case, the leaf space $\underline{\mathcal{M}}$ of the Reeb foliation locally inherits an almost Kähler structure $(\underline{h},\underline{J})$ . In addition,

•

$s$ satisfies (2.16b) if and only if the Ricci tensor of $\underline{h}$ commutes with $\underline{J}$ ;
•

$s$ satisfies (2.16b) and (2.16c) if and only if $\underline{h}$ is Einstein.

In general, a CR–Einstein structure does not arise from a density $\sigma$ of weight $(1,0)$ that satisfies (2.15a) as illustrated by the example in Appendix A. It is nonetheless instructive to write down a system of equations describing such a structure.

Suppose that an almost CR manifold $(\mathcal{M},H,J)$ admits a density $\sigma$ of weight $(1,0)$ that satisfies


	$\displaystyle\nabla_{\bar{\alpha}}\sigma=0\,,$		(2.17a)
	$\displaystyle\nabla_{(\alpha}\nabla_{\beta)}\sigma+\mathrm{i}\mathsf{A}_{\alpha\beta}\sigma+\frac{1}{m}\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}\sigma=0\,,$		(2.17b)
	$\displaystyle\nabla_{\gamma}\sigma\mathsf{N}^{\gamma}{}_{(\bar{\alpha}\bar{\beta})}+\frac{1}{m}\nabla_{\gamma}\mathsf{N}^{\gamma}{}_{(\bar{\alpha}\bar{\beta})}\sigma=0\,,$		(2.17c)
	$\displaystyle\left(2\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}-\mathsf{N}_{\gamma\delta\alpha}\mathsf{N}^{\gamma\delta}{}_{\bar{\beta}}\right)_{\circ}=0\,.$		(2.17d)

Then the $s:=\sigma\overline{\sigma}$ is an almost CR–Einstein scale.

It is a simple matter of setting $s=\sigma\overline{\sigma}$ , substituting this into equations (2.16) and applying (2.17a) repeatedly. This directly yields equations (2.17b) and (2.17c), while equation (2.16c) reduces to

\displaystyle\left(\nabla_{\bar{\beta}}\nabla_{\alpha}\sigma+\mathsf{P}_{\alpha\bar{\beta}}\sigma-\frac{1}{m+2}\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}\sigma\right)_{\circ}=0\,,

which, upon (2.4a), can be shown to be equivalent to (2.17d). ∎

As pointed out in [Mat22], the system of equations (2.17a) and (2.17b) arise from the first BGG operator on a density $\sigma\in\Gamma(\mathcal{E}(1,0))$ given by

\displaystyle\sigma\mapsto\left(\nabla_{\bar{\alpha}}\sigma,\nabla_{(\alpha}\nabla_{\beta)}\sigma+\mathrm{i}\mathsf{A}_{\alpha\beta}\sigma+\frac{1}{m}\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}\sigma\right)\,.

A related CR invariant equation of interest is given by, for $\sigma\in\Gamma(\mathcal{E}(1,w))$ for any $w\in\mathbf{Z}$ ,

\displaystyle\nabla_{(\alpha}\nabla_{\beta)}\sigma+\mathrm{i}\mathsf{A}_{\alpha\beta}\sigma+\frac{1}{m}\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}\sigma

\displaystyle=0\,,

or equivalently,

\displaystyle\nabla_{\alpha}\nabla_{\beta}s+\mathrm{i}\mathsf{A}_{\alpha\beta}s+\frac{1}{m}\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}s+\mathsf{N}_{\gamma[\alpha\beta]}\nabla^{\gamma}s

\displaystyle=0\,.

If $\sigma$ is a real density of weight $(1,1)$ , then the CR scale determines a contact form, whose pseudo-Hermitian invariants satisfy

\displaystyle\mathsf{A}_{\alpha\beta}

\displaystyle=\frac{1}{m}\mathrm{i}\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}\,.

Interestingly, solutions (2.17) provide a nice interpretation of $\|\mathsf{N}\|^{-2}$ in the following terms.

Let $(\mathcal{M},H,J)$ be an almost CR manifold. Assume that $\|\mathsf{N}\|^{2}$ nowhere vanishes. Suppose it admits a solution $\sigma\in\Gamma(\mathcal{E}(1,0))$ to (2.17). Then the almost CR–Einstein scale $s:=\sigma\overline{\sigma}$ is proportional to $\|\mathsf{N}\|^{-2}$ by a constant factor.

Suppose that equations (2.17) hold. Then the commutation relations (2.4), where $\nabla$ is taken to be the Webster connection preserving $s$ , reduce to


$\displaystyle 0$	$\displaystyle=\frac{1}{m+2}\left(-\mathsf{N}_{\gamma\delta\alpha}\mathsf{N}^{\gamma\delta}{}_{\bar{\beta}}+2\mathsf{N}_{\alpha\gamma\delta}{\mathsf{N}}_{\bar{\beta}}{}^{\gamma\delta}\right)\sigma+h_{\alpha\bar{\beta}}\left(\frac{\Lambda}{m+2}\sigma-\mathrm{i}{\nabla}_{0}\sigma\right)\,,$	(2.18a)
$\displaystyle{\nabla}_{\alpha}{\nabla}_{0}\sigma$	$\displaystyle=0\,,$	(2.18b)
$\displaystyle 0$	$\displaystyle=\frac{1}{m+2}{\nabla}^{\gamma}{\mathsf{N}}_{\alpha\beta\gamma}\sigma\,.$	(2.18c)

Contracting equation (2.18a) gives $\|\mathsf{N}\|^{2}=-m\Lambda+\mathrm{i}m(m+2)\sigma^{-1}{\nabla}_{0}\sigma$ . Hence, using equation (2.18b), we find that $\nabla_{\alpha}\left(\|\mathsf{N}\|^{2}\right)=0$ , and using the commutation once more, we find that $\|\mathsf{N}\|^{2}$ is constant, and so its inverse must be proportional to $s$ by a constant factor. ∎

With reference to the above result, since $\Lambda$ as defined by (2.13) is constant, $\mathsf{Sc}$ must also be constant with respect to the Webster connection $\nabla$ compatible with $\|\mathsf{N}\|^{2}$ . Note also that equation (2.18c) tells us that ${\nabla}^{\gamma}{\mathsf{N}}_{\gamma\alpha\beta}=0$ whenever $\sigma\neq 0$ .

Throughout $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}})$ will denote an oriented and time-oriented conformal manifold of dimension $n+2$ and of Lorentzian signature $(+,\ldots,+,-)$ . For consistency with the rest of the article, we shall adorn tensors on $\widetilde{\mathcal{M}}$ with a tilde. Two metrics $\widetilde{g}$ and $\widehat{\widetilde{g}}$ belong to the conformal class $\widetilde{\mathbf{c}}$ if and only if

\displaystyle\widehat{\widetilde{g}}

\displaystyle=\mathrm{e}^{2\,\widetilde{\varphi}}\widetilde{g}\,,

for some smooth function

\widetilde{\varphi}

\widetilde{\mathcal{M}}

(3.1)

Following [BEG94], for each $w\in\mathbf{R}$ , there are naturally associated density bundles denoted $\widetilde{\mathcal{E}}[w]$ , and the Levi-Civita connection of $\widetilde{g}$ extends to a linear connection on $\widetilde{\mathcal{E}}[w]$ . In particular, metrics in $\widetilde{\mathbf{c}}$ are in one-to-one correspondence with sections of the bundle of conformal scales, denoted $\widetilde{\mathcal{E}}_{+}[1]$ , which is a choice of square root of $\widetilde{\mathcal{E}}[2]$ . The correspondence is achieved by means of the conformal metric $\widetilde{\bm{g}}$ , which is a distinguished non-degenerate section of $\scaleobj{1.2}{\odot}^{2}T\widetilde{\mathcal{M}}\otimes\widetilde{\mathcal{E}}[2]$ with the property that if $\widetilde{\sigma}\in\widetilde{\mathcal{E}}[1]$ is a conformal scale, then the corresponding metric in $\widetilde{\mathbf{c}}$ is given by $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ . The Levi-Civita connection $\widetilde{\nabla}$ of $\widetilde{g}$ also preserves $\widetilde{\sigma}$ , and it follows that $\widetilde{\bm{g}}$ is preserved by the Levi-Civita connection of any metric $\widetilde{g}$ in $\widetilde{\mathbf{c}}$ . We thus have a canonical identification of $T\widetilde{\mathcal{M}}$ with $T^{*}\widetilde{\mathcal{M}}\otimes\widetilde{\mathcal{E}}[2]$ via $\widetilde{\bm{g}}$ .

We shall often use the abstract index notation, whereby sections of $T\widetilde{\mathcal{M}}$ , respectively, $T^{*}\widetilde{\mathcal{M}}$ will be adorned with upper, respectively, lower minuscule Roman indices starting from the beginning of the alphabet, e.g. $\widetilde{v}^{a}\in\Gamma(T\widetilde{\mathcal{M}})$ , and $\widetilde{\xi}_{ab}\in\Gamma(\scaleobj{1.2}{\wedge}^{2}T^{*}\widetilde{\mathcal{M}})$ . Symmetrisation will be denoted by round brackets, and skew-symmetrisation by square brackets as before. Indices will be lowered and raised with $\widetilde{\bm{g}}_{ab}$ and its inverse $\widetilde{\bm{g}}^{ab}$ respectively. The tracefree part of symmetric tensors will be denoted by a ring.

For any two metrics $\widetilde{g}$ and $\widehat{\widetilde{g}}$ in $\widetilde{\mathbf{c}}$ related by (3.1), their respective Levi-Civita connections $\widetilde{\nabla}$ and $\widehat{\widetilde{\nabla}}$ are related by

\displaystyle\widehat{\widetilde{\nabla}}_{a}\widetilde{\alpha}_{b}

\displaystyle=\widetilde{\nabla}_{a}\widetilde{\alpha}_{b}+(w-1)\widetilde{\Upsilon}_{a}\widetilde{\alpha}_{b}-\widetilde{\Upsilon}_{b}\widetilde{\alpha}_{a}+\widetilde{\Upsilon}_{c}\widetilde{\alpha}^{c}\widetilde{\bm{g}}_{ab}\,,

\displaystyle\widetilde{\alpha}_{a}\in\Gamma(\widetilde{\mathcal{E}}_{a}[w])\,,

where $\widetilde{\Upsilon}_{a}:=\widetilde{\nabla}_{a}\widetilde{\varphi}$ .

By convention, we take the Riemann tensor of a given metric $\widetilde{g}_{ab}$ in $\widetilde{\mathbf{c}}$ to be defined by

\displaystyle 2\widetilde{\nabla}_{[a}\widetilde{\nabla}_{b]}\widetilde{\alpha}_{c}

\displaystyle=-\widetilde{\mathsf{R}}_{ab}{}^{d}{}_{d}\widetilde{\alpha}_{d}\,,

\displaystyle\widetilde{\alpha}_{a}\in\Gamma(\widetilde{\mathcal{E}}_{a}[w])\,.

The Riemann tensor decomposes as

\displaystyle\widetilde{\mathsf{R}}_{abcd}

\displaystyle=\widetilde{\mathsf{W}}_{abcd}+4\widetilde{\bm{g}}_{[a|[c}\widetilde{\mathsf{P}}_{d]|b]}\,,

where $\widetilde{\mathsf{W}}_{abcd}$ is the Weyl tensor and $\widetilde{\mathsf{P}}_{ab}$ the Schouten tensor, which is related to the Ricci tensor $\widetilde{\mathsf{Ric}}_{ab}=\widetilde{\mathsf{R}}_{ca}{}^{c}{}_{b}$ and the Ricci scalar $\widetilde{\mathsf{Sc}}=\widetilde{\mathsf{Ric}}_{a}{}^{a}$ by

\displaystyle\widetilde{\mathsf{P}}_{ab}

\displaystyle=\frac{1}{n}\left(\widetilde{\mathsf{Ric}}_{ab}-\frac{\widetilde{\mathsf{Sc}}}{2(n+1)}\widetilde{g}_{ab}\right)\,.

The Schouten scalar is defined to be $\widetilde{\mathsf{P}}:=\widetilde{\mathsf{P}}_{ab}\widetilde{\bm{g}}^{ab}=\frac{1}{2(n+1)}\widetilde{\mathsf{Sc}}$ . The Cotton tensor is given by

\displaystyle\widetilde{\mathsf{Y}}_{cab}

\displaystyle=2\widetilde{\nabla}_{[a}\widetilde{\mathsf{P}}_{b]c}\,,

and, by the Bianchi identities, satisfies $(n-1)\widetilde{\mathsf{Y}}_{cab}=\widetilde{\nabla}^{d}\widetilde{\mathsf{W}}_{dcab}$ .

While the Weyl tensor is conformally invariant, the Schouten tensor, Schouten scalar and Cotton tensor transform as

	$\displaystyle\widehat{\widetilde{\mathsf{P}}}_{ab}$	$\displaystyle=\widetilde{\mathsf{P}}_{ab}-\widetilde{\nabla}_{a}\widetilde{\Upsilon}_{b}+\widetilde{\Upsilon}_{a}\widetilde{\Upsilon}_{b}-\frac{1}{2}\widetilde{\Upsilon}^{c}\widetilde{\Upsilon}_{c}\widetilde{\bm{g}}_{ab}\,,$	$\displaystyle\widehat{\widetilde{\mathsf{P}}}$	$\displaystyle=\widetilde{\mathsf{P}}-\widetilde{\nabla}^{a}\widetilde{\Upsilon}_{a}-\frac{n}{2}\widetilde{\Upsilon}^{c}\widetilde{\Upsilon}_{c}\,,$		(3.2)
	$\displaystyle\widehat{\widetilde{\mathsf{Y}}}_{cab}$	$\displaystyle=\widetilde{\mathsf{Y}}_{cab}+\widetilde{\Upsilon}^{d}\widetilde{\mathsf{W}}_{dcab}\,,$				(3.3)

respectively.

We recall a number of notions introduced in \citesTrautman1984,Trautman1985,Robinson1985,Penrose1986,Robinson1986,Robinson1989,Trautman1999,Fino2020. An optical geometry consists of a triple $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ , where $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}})$ is an oriented and time-oriented Lorentzian conformal manifold of dimension $n+2$ and $\widetilde{K}$ is a null line distribution, which we shall also referred to as an optical structure. The rank- $n$ screen bundle $\widetilde{H}:=\widetilde{K}^{\perp}/\widetilde{K}$ inherits a conformal structure $\widetilde{\mathbf{c}}_{\widetilde{H}}$ of Riemannian signature.

Of importance in the present article is when $\widetilde{K}$ is tangent to a non-shearing congruence of null geodesics $\widetilde{\mathcal{K}}$ , i.e. for any non-vanishing section $\widetilde{k}$ of $\widetilde{K}$ , i.e.

\displaystyle\mathsterling_{\widetilde{k}}\widetilde{g}(\widetilde{v},\widetilde{w})

\displaystyle=\widetilde{\epsilon}\widetilde{g}(\widetilde{v},\widetilde{w})\,,

\displaystyle\widetilde{v},\widetilde{w}\in\Gamma(\widetilde{K}^{\perp})\,,

(3.4)

for some smooth function $\widetilde{\epsilon}$ . This means that the integral curves of $\widetilde{k}$ are null geodesics, and the conformal structure on $\widetilde{H}$ is preserved along these. The local leaf space $\mathcal{M}$ of $\widetilde{\mathcal{K}}$ thus inherits a rank- $n$ distribution $H$ from $\widetilde{H}$ , equipped with a bundle conformal structure of Riemannian signature from $\widetilde{\mathbf{c}}_{\widetilde{H}}$ .

In addition, we shall assume that $\widetilde{\mathcal{K}}$ is twisting, i.e. for any one-form $\widetilde{\kappa}\in\Gamma(\mathrm{Ann}(\widetilde{H}))$ ,

\displaystyle\mathrm{d}\widetilde{\kappa}(\widetilde{v},\widetilde{w})

\displaystyle\neq 0\,,

\displaystyle\widetilde{v},\widetilde{w}

\displaystyle\in\Gamma(\widetilde{K}^{\perp})\,,

(3.5)

which means that $\widetilde{K}^{\perp}$ , and thus $H$ , are not integrable.

When $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}})$ is of dimension $2m+2$ , a particular case of an optical structure is provided by the notion of almost Robinson structure \citesNurowski2002,Fino2023, that is, a pair $(\widetilde{N},\widetilde{K})$ where $\widetilde{N}$ is a totally null complex $(m+1)$ -plane distribution, i.e.

\displaystyle\widetilde{\bm{g}}(\widetilde{v},\widetilde{w})

\displaystyle=0\,,

for all

\widetilde{v},\widetilde{w}\in\Gamma(\widetilde{N})

and $\widetilde{K}$ a real null line distribution such that $\mathbf{C}\otimes\widetilde{K}=\widetilde{N}\cap\overline{\widetilde{N}}$ . One can show that $(\widetilde{N},\widetilde{K})$ is equivalent to an optical structure $\widetilde{K}$ whose screen bundle is equipped with a bundle complex structure $\widetilde{J}$ compatible with the induced conformal structure $\widetilde{\mathbf{c}}_{\widetilde{H}}$ . Again, let us consider the leaf space $\mathcal{M}$ of the congruence $\widetilde{\mathcal{K}}$ of null curves tangent to $\widetilde{K}$ . When $\widetilde{N}$ is preserved along the flow of any generator of $\widetilde{\mathcal{K}}$ , i.e. $[\widetilde{K},\widetilde{N}]\subset\widetilde{N}$ , we refer to $(\widetilde{N},\widetilde{K})$ as a nearly Robinson structure. In this case, the curves of $\widetilde{K}$ are geodesics, and the leaf space $\mathcal{M}$ inherits an almost CR structure $(H,J)$ . If in addition, $\widetilde{N}$ is involutive, i.e. $[\widetilde{N},\widetilde{N}]\subset\widetilde{N}$ , we refer to $(\widetilde{N},\widetilde{K})$ simply as a Robinson structure, which also implies the involutivity of the almost CR structure $(H,J)$ on $\mathcal{M}$ [NT02, FLTC23].

In dimension four, an optical structure is equivalent to an almost Robinson structure. The former is tangent to a non-shearing congruence of null geodesics if and only if the latter is involutive. This being case, the Weyl tensor satisfies the integrability condition

\displaystyle\widetilde{W}(\widetilde{k},\widetilde{v},\widetilde{k},\widetilde{v})

\displaystyle=0\,,

for any

\widetilde{k}\in\Gamma(\widetilde{K})

\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})

(3.6)

In even dimensions greater than four, however, an optical geometry does not a priori admit a distinguished almost Robinson structure. We note however the following special case: we say that an almost Robinson structure $(\widetilde{N},\widetilde{K})$ is twist-induced if the congruence $\widetilde{\mathcal{K}}$ tangent to $\widetilde{K}$ is geodesic and twisting, and the associated bundle complex structure $\widetilde{J}$ on the screen bundle $\widetilde{H}$ is compatible with the twist in the sense that, for any non-vanishing section $\widetilde{\kappa}\in\Gamma(\widetilde{K}^{\perp})$ and any bundle metric $\widetilde{h}$ in $\widetilde{\mathbf{c}}_{\widetilde{H}}$ ,

\displaystyle\mathrm{d}\widetilde{\kappa}(\widetilde{v},\widetilde{w})

\displaystyle\propto\widetilde{h}\left(\widetilde{J}(\widetilde{v}+\widetilde{K}),\widetilde{w}+\widetilde{K}\right)\,,

for all

\widetilde{v},\widetilde{w}\in\Gamma(\widetilde{N})

If in addition, $\widetilde{\mathcal{K}}$ is non-shearing, then $(\widetilde{N},\widetilde{K})$ descends to a partially integrable contact almost CR structure of positive definite signature on the leaf space $\mathcal{M}$ of $\widetilde{\mathcal{K}}$ .

In even higher dimensions, we have the next result:

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ be an optical geometry of dimension $2m+2$ equipped with a twisting non-shearing congruence of null geodesics $\widetilde{\mathcal{K}}$ . Then the twist of $\widetilde{\mathcal{K}}$ induces a nearly Robinson structure $(\widetilde{N},\widetilde{K})$ if and only if the Weyl tensor satisfies (3.6). This being the case, the local leaf space $\mathcal{M}$ of the congruence inherits a partially integrable contact almost CR structure $(H,J)$ from $(\widetilde{N},\widetilde{K})$ .

For $m=1$ , the above theorem is perfectly consistent with the extent literature since the integrability condition then becomes vacuous.

Let us focus on a twist-induced nearly Robinson geometry $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{K})\rightarrow(\mathcal{M},H,J)$ with non-shearing congruence. For a congruence of null curves, the geodesic property of $\widetilde{K}$ , its shear and twist are all conformally invariant and do not depend on the choice of generator of $\widetilde{K}$ . The question that we need to address is how metrics in $\widetilde{\mathbf{c}}$ are related to contact forms for $(\mathcal{M},H,J)$ . To this end, we shall use the following canonical structures arising from our geometric setup [FLTC20, FLTC23, TC22]:

•

There is a conformal subclass $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ of metrics in $\widetilde{\mathbf{c}}$ with the property that whenever $\widetilde{g}$ is in $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ , the congruence $\widetilde{\mathcal{K}}$ is non-expanding, i.e.

\displaystyle\mathsterling_{\widetilde{k}}\widetilde{g}(\widetilde{v},\widetilde{w})

\displaystyle=0\,,

\displaystyle\widetilde{v},\widetilde{w}

\displaystyle\in\Gamma(\widetilde{K}^{\perp})\,,

(3.7)

Any two metrics in $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ differ by a factor constant along $\widetilde{K}$ .

•

From the twisting property of $\widetilde{\mathcal{K}}$ , there is a distinguished generator $\widetilde{k}$ of $\widetilde{\mathcal{K}}$ with the property that for any metric $\widetilde{g}$ in $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ , we have that

$\displaystyle\widetilde{\kappa}$ $\displaystyle=\widetilde{g}(\widetilde{k},\cdot)=2\theta\,,$

where $\theta$ is a contact form for $(H,J)$

Thanks to $\widetilde{k}$ and $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ , we obtain a ‘tight’ relation between $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ and the leaf space $(\mathcal{M},H,J)$ of $\widetilde{\mathcal{K}}$ . To be precise, there is a one-to-one correspondence between metrics in $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ and contact forms for $(H,J)$ . Further, each metric $\widetilde{g}_{\theta}$ in $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ associated to some contact form $\theta$ takes the form

for some complex-valued functions $\widetilde{\lambda}_{\alpha}$ and $\widetilde{\lambda}_{\bar{\alpha}}=\overline{\widetilde{\lambda}_{\alpha}}$ , and real function $\widetilde{\lambda}_{0}$ on on $\widetilde{\mathcal{M}}$ . Differentiation with respect to $\phi$ will be denoted by a dot, i.e. $\dot{\widetilde{f}}:=\mathsterling_{\widetilde{k}}\widetilde{f}$ for any smooth tensor-valued function $\widetilde{f}$ on $\widetilde{\mathcal{M}}$ , and this notation will be extended to tensor components.

Any other choice of metric $\widetilde{g}{}_{\widehat{\theta}}=\mathrm{e}^{\varphi}\widetilde{g}{}_{\theta}$ in $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ for some smooth function $\varphi$ on $\mathcal{M}$ is given by

\displaystyle\widetilde{g}_{\widehat{\theta}}

\displaystyle=4\widehat{\theta}\odot\widehat{\widetilde{\lambda}}+\widehat{h}\,,

where

	$\displaystyle\widehat{\theta}$	$\displaystyle=\frac{1}{2}\widetilde{g}_{\widehat{\theta}}(\widetilde{k},\cdot)=\mathrm{e}^{\varphi}\theta\,,$
	$\displaystyle\widehat{h}$	$\displaystyle=\mathrm{e}^{\varphi}\left(h-2\mathrm{i}\Upsilon_{\alpha}\theta^{\alpha}\odot\theta+2\mathrm{i}\Upsilon_{\bar{\alpha}}\overline{\theta}^{\bar{\alpha}}\odot\theta+2\Upsilon_{\alpha}\Upsilon^{\alpha}\theta\odot\theta\right)\,,$
	$\displaystyle\widehat{\widetilde{\lambda}}$	$\displaystyle=\widetilde{\lambda}+\frac{1}{2}\mathrm{i}\Upsilon_{\alpha}\theta^{\alpha}-\frac{1}{2}\mathrm{i}\Upsilon_{\bar{\alpha}}\overline{\theta}{}^{\bar{\alpha}}-\frac{1}{2}\Upsilon_{\alpha}\Upsilon^{\alpha}\theta\,,$

with $(\theta^{\alpha})$ being admissible for $(H,J,\theta)$ , and $\Upsilon_{\alpha}=\nabla_{\alpha}\varphi$ . On the other hand, a change of affine parametrisation

\displaystyle\phi\mapsto\phi^{\prime}=\phi-\mathring{\phi}\,,

(3.9)

for some smooth function $\mathring{\phi}$ on $\mathcal{M}$ accompanied with the redefinition

\displaystyle\widetilde{\lambda}^{\prime}_{\alpha}

\displaystyle=\widetilde{\lambda}_{\alpha}+\nabla_{\alpha}\mathring{\phi}\,,

\displaystyle\widetilde{\lambda}^{\prime}_{0}

\displaystyle=\widetilde{\lambda}_{0}+\nabla_{0}\mathring{\phi}\,,

preserves the form of the metric (3.8) and clearly the vector field $\widetilde{k}=\frac{\partial}{\partial\phi^{\prime}}$ .

In the light of Theorem 3.1, we shall henceforth denote a twist-induced nearly Robinson geometry with non-shearing congruence by the quadruple $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})\longrightarrow(\mathcal{M},H,J)$ where $\widetilde{k}$ is the distinguished section of $\widetilde{K}$ as described above. The perturbed Fefferman spaces to be introduced in Section 4 are a special case of these.

Recall that a metric $\widetilde{g}$ in $\widetilde{\mathbf{c}}$ is said to be Einstein if its Ricci tensor satisfies

\displaystyle\widetilde{\mathsf{Ric}}

\displaystyle=\widetilde{\Lambda}\widetilde{g}\,,

for some constant

\widetilde{\Lambda}

In the context of optical geometries, one can make further definitions whereby the Einstein condition is weakened.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{K})$ be an almost Robinson geometry. We say that a metric $\widetilde{g}$ in $\widetilde{\mathbf{c}}$ is:

•

a weakly half-Einstein metric if its Ricci tensor satisfies

\displaystyle\widetilde{\mathsf{Ric}}_{ab}\widetilde{v}^{a}\widetilde{v}^{b}

\displaystyle=0\,,

for any

\widetilde{v}\in\Gamma(\widetilde{N})

;

(3.10)

•

a half-Einstein metric if it is weakly half-Einstein and has constant Ricci scalar curvature.

In the assumption of an optical geometry only, we have the following.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ be an optical geometry. A metric $\widetilde{g}$ in $\widetilde{\mathbf{c}}$ is said to be a pure radiation metric if its Ricci scalar is constant, and the tracefree part of the Ricci tensor satisfies

\displaystyle\widetilde{\mathsf{Ric}}_{ab}\widetilde{v}^{a}

\displaystyle=\widetilde{\Lambda}\widetilde{g}_{ab}\widetilde{v}^{a}\,,

for any

\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})

;

(3.11)

Clearly, if $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{K})$ is an almost Robinson geometry that admits a pure radiation metric $\widetilde{g}$ in $\widetilde{\mathbf{c}}$ , then $\widetilde{g}$ is in particular, (weakly) half-Einstein.

For our considerations, these conditions are somewhat too strong. For this reason, and motivated by the notion of almost pseudo-Riemannian structure in [CG18], we make the following definition:

Let $\widetilde{\sigma}\in\Gamma(\widetilde{\mathcal{E}}[1])$ with zero set $\widetilde{\mathcal{Z}}=\{\widetilde{p}\in\widetilde{\mathcal{M}}|\widetilde{\sigma}(\widetilde{p})=0\}$ . We say that $\widetilde{\sigma}$ is an almost Lorentzian scale if $\widetilde{\nabla}\widetilde{\sigma}\neq 0$ on $\widetilde{\mathcal{Z}}$ where $\widetilde{\nabla}$ is the Levi-Civita connection of some (and thus any) metric in $\widetilde{\mathbf{c}}$ .

That this definition does not depend on the choice of metric is easy to check. The density $\widetilde{\sigma}$ defines a metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ in $\widetilde{\mathbf{c}}$ , but regular only off $\widetilde{\mathcal{Z}}$ .

First introduced by Gover in [Gov05] — see also [LeB85] — an almost Einstein scale is an almost Lorentzian scale $\widetilde{\sigma}$ that satisfies the conformally invariant equation

\displaystyle\left(\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\widetilde{\sigma}+\widetilde{\mathsf{P}}_{ab}\widetilde{\sigma}\right)_{\circ}=0\,.

(3.12)

If $\widetilde{\sigma}$ has empty zero set $\widetilde{\mathcal{Z}}$ , it is then referred to as an Einstein scale, and it defines a (global) Einstein metric. Otherwise, if $\widetilde{\mathcal{Z}}$ is non-empty, then off $\widetilde{\mathcal{Z}}$ , the metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ is Einstein. A useful interpretation of $\widetilde{\mathcal{Z}}$ is as the conformal infinity of some Lorentzian Einstein manifold — see [CG18] for more details.

We can readily generalise this idea to the metrics introduced in Definition 3.3.

Let $\widetilde{\sigma}$ be an almost Lorentzian scale on an almost Robinson geometry $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{K})$ . We say that $\widetilde{\sigma}$ is

•

an almost weakly half-Einstein scale if it satisfies


		$\displaystyle\left(\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\widetilde{\sigma}+\widetilde{\mathsf{P}}_{ab}\widetilde{\sigma}\right)_{\circ}=\frac{1}{2}\widetilde{\Phi}_{ab}\widetilde{\sigma}\,,$		(3.13a)
for some tracefree symmetric tensor $\widetilde{\Phi}_{ab}$ satisfying

	$\displaystyle\widetilde{\Phi}_{ab}\widetilde{v}^{a}\widetilde{v}^{b}$	$\displaystyle=0\,,$	for any $\widetilde{v}^{a}\in\Gamma(\widetilde{N})$ ;	(3.13b)

•

an almost half-Einstein scale if it is an almost weakly half-Einstein scale, and

\displaystyle\widetilde{\Phi}_{a}{}^{b}\widetilde{\nabla}_{b}\widetilde{\sigma}-\frac{1}{2}\widetilde{\sigma}\widetilde{\nabla}_{b}\widetilde{\Phi}_{a}{}^{b}=0\,.

(3.14)

Similarly, we introduce the following.

Let $\widetilde{\sigma}$ be an almost Lorentzian scale on an optical geometry $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ . We say that $\widetilde{\sigma}$ is an almost pure radiation scale if it satisfies

\displaystyle\left(\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\widetilde{\sigma}+\widetilde{\mathsf{P}}_{ab}\widetilde{\sigma}\right)_{\circ}=\frac{1}{2}\widetilde{\Phi}_{ab}\widetilde{\sigma}\,,

\displaystyle\widetilde{\Phi}_{a}{}^{b}\widetilde{\nabla}_{b}\widetilde{\sigma}-\frac{1}{2}\widetilde{\sigma}\widetilde{\nabla}_{b}\widetilde{\Phi}_{a}{}^{b}=0\,,

(3.15)

where

\displaystyle\widetilde{\Phi}_{ab}\widetilde{v}^{b}=0\,,

for any

\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})

(3.16)

All the conditions given in the above definition are conformally invariant.

Note that condition (3.16) is equivalent to

\displaystyle\widetilde{\Phi}_{ab}

\displaystyle=\widetilde{\bm{\Phi}}\widetilde{\bm{\kappa}}_{a}\widetilde{\bm{\kappa}}_{b}\,,

for some

\widetilde{\bm{\Phi}}\in\Gamma(\widetilde{\mathcal{E}}[-4])

(3.17)

The relation between Definition 3.3 and Definition 3.7 is given below.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{K})$ be an almost Robinson geometry, and let $\widetilde{\sigma}\in\Gamma(\mathcal{E}[1])$ so that $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}\in\widetilde{\mathbf{c}}$ is a smooth metric off the zero set of $\widetilde{\sigma}$ . Then

(1)

$\widetilde{\sigma}$ is an almost weakly half-Einstein scale if and only if $\widetilde{g}$ is a weakly half-Einstein metric;
(2)

$\widetilde{\sigma}$ is an almost half-Einstein scale if and only if $\widetilde{g}$ is a half-Einstein metric.

Throughout $\widetilde{\nabla}{}^{(\widetilde{g})}$ will denote the Levi-Civita connection preserving $\widetilde{g}$ , so that for any other connection $\widetilde{\nabla}$ , $\widetilde{\nabla}=\widetilde{\nabla}{}^{(\widetilde{g})}+\Upsilon$ where $\widetilde{\Upsilon}=\widetilde{\sigma}^{-1}\widetilde{\nabla}\widetilde{\sigma}$ . We first prove the implication $\Rightarrow$ for each of the cases, working off the zero set of $\widetilde{\sigma}$ :

(1)

Suppose $\widetilde{\sigma}$ satisfies (3.13). Multiplying (3.13) through by $\widetilde{\sigma}^{-1}$ and using the transformation rule for the Schouten tensor (3.2), we find $\left(\widetilde{\mathsf{Ric}}{}^{(\widetilde{g})}_{ab}\right)_{\circ}=\frac{1}{2}\widetilde{\Phi}_{ab}$ , and comparing (3.10) with (3.13b) tells us that $\widetilde{g}$ is a weakly half-Einstein metric.

(2)

Suppose that $\widetilde{\sigma}$ satisfies (3.14) in addition to (3.13). By case (1), we already know $\widetilde{g}$ is weakly half-Einstein metric. We proceed to check that its scalar curvature is constant. Multiplying (3.14) through by $\widetilde{\sigma}^{-1}$ gives the transformation rule

\displaystyle\widetilde{\nabla}{}^{(\widetilde{g})}_{b}\widetilde{\Phi}_{a}{}^{b}

\displaystyle=\widetilde{\nabla}_{b}\widetilde{\Phi}_{a}{}^{b}+2\widetilde{\Upsilon}{}_{b}\widetilde{\Phi}_{a}{}^{b}\,.

The right-hand-side is zero by virtue of (3.14). On the other hand, we have $(\widetilde{\mathsf{Ric}}{}^{(\widetilde{g})}_{ab})_{\circ}=\widetilde{\Phi}_{ab}$ , and the contracted Bianchi identity tells us that

\displaystyle\widetilde{\nabla}_{a}\widetilde{\mathsf{Sc}}{}^{(\widetilde{g})}

\displaystyle=4\widetilde{\nabla}{}^{(\widetilde{g})}_{b}\widetilde{\Phi}_{a}{}^{b}=0\,,

i.e. $\widetilde{\mathsf{Sc}}{}^{(\widetilde{g})}$ is constant, which makes $\widetilde{g}$ a half-Einstein metric.

For the converse direction, if we assume that the metric $\widetilde{g}$ defined by $\widetilde{\sigma}$ is weakly half-Einstein off the zero set $\widetilde{\mathcal{Z}}$ of $\widetilde{\sigma}$ , then choosing $\widetilde{\nabla}$ to preserve $\widetilde{\sigma}$ tells us that (3.13) is satisfied off $\widetilde{\mathcal{Z}}$ . But (3.13) clearly holds on $\widetilde{\mathcal{Z}}$ too. Thus $\widetilde{\sigma}$ is an almost weakly half-Einstein scale. The almost half-Einstein case is similar. ∎

In a completely analogous way, we prove:

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ be an optical geometry, and let $\widetilde{\sigma}\in\Gamma(\mathcal{E}[1])$ so that $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}\in\widetilde{\mathbf{c}}$ is a smooth metric off the zero set of $\widetilde{\sigma}$ . Then $\widetilde{\sigma}$ is an almost pure radiation scale if and only if $\widetilde{g}$ is a pure radiation metric.

Following from the proof of Proposition 3.10, suppose that $\widetilde{\Phi}_{ab}$ satisfies (3.16), in addition to $\widetilde{\sigma}$ being a solution to (3.14). Then clearly (3.11) follows from (3.16). The converse works in the same way as in the proof of Proposition 3.10. ∎

Let $(\mathcal{M},H,J)$ be almost CR manifold of dimension $2m+1$ . Henceforth, we assume that the Levi form of $(H,J)$ is positive definite, although this requirement may be relaxed with caution. We adapt the approach of [ČG08] and [Lei07]. The bundle $\mathcal{E}(-1,0)$ with its zero-section removed is a principal bundle with structure group $\mathbf{C}^{*}$ , and taking its quotient by the natural $\mathbf{R}_{>0}$ -action yields a circle bundle $\widetilde{\mathcal{M}}$ over $\mathcal{M}$ . The projection from $\mathcal{E}(-1,0)$ with its zero section removed to $\widetilde{\mathcal{M}}$ sends a non-vanishing density $\tau$ of weight $(-1,0)$ to an equivalence class $[\tau]$ , where $\tau,\tau^{\prime}\in[\tau]$ if and only if $\tau=\varrho\tau^{\prime}$ for some positive real function $\varrho$ on $\mathcal{M}$ . Let us fix a pseudo-Hermitian structure $\theta$ . We then have a natural identification of sections of $\widetilde{\mathcal{M}}$ with densities $\tau$ of weight $(-1,0)$ satisfying $\theta=\tau\overline{\tau}\bm{\theta}$ , and we may define a fibre coordinate $\phi\in[-\pi,\pi)$ such that $\mathrm{e}^{\mathrm{i}\phi}\tau$ is a section of $\widetilde{\mathcal{M}}\rightarrow\mathcal{M}$ . For each $\alpha\in\mathbf{R}$ , we can then define the Lorentzian metric on $\widetilde{\mathcal{M}}$ given by

\displaystyle\widetilde{g}_{\theta}^{(\alpha)}

\displaystyle=4\theta\odot\left(\mathrm{d}\phi+\frac{\mathrm{i}}{2}\left(\sigma^{-1}\nabla\sigma-\overline{\sigma}^{-1}\nabla\overline{\sigma}\right)-\left(\frac{1}{m+2}\mathsf{P}+\frac{\alpha}{2m(m+1)}\|\mathsf{N}\|^{2}\right)\theta\right)+h\,,

(4.1)

where $\sigma=\tau^{-1}$ , $h$ is (the degenerate metric induced by) the Levi form of $\theta$ , $\mathsf{P}$ is the Websten–Schouten scalar of $\theta$ and $\|\mathsf{N}\|^{2}$ is the norm squared of the Nijenhuis tensor of $\theta$ with respect to $h$ .

We shall refer to the metric defined in (4.1) as the $\alpha$ -Fefferman metric associated to $\theta$ .

One can conveniently eliminate $\overline{\sigma}$ by noting that since $\sigma$ determines $\nabla$ , we have that $\nabla(\sigma\overline{\sigma})=0$ , i.e. $\sigma^{-1}\nabla\sigma=-\overline{\sigma}^{-1}\nabla\overline{\sigma}$ . Using (2.3), we can also easily recover the more familiar form of the Fefferman metric in terms of the connection one-form:

\displaystyle\widetilde{g}_{\theta}^{(\alpha)}

\displaystyle=4\theta\odot\left(\mathrm{d}\phi+\frac{1}{m+2}\left(\mathrm{i}\Gamma_{\alpha}{}^{\alpha}-\frac{\mathrm{i}}{2}h^{\alpha\bar{\beta}}\mathrm{d}h_{\alpha\bar{\beta}}-\left(\mathsf{P}+\frac{\alpha(m+2)}{2m(m+1)}\|\mathsf{N}\|^{2}\right)\theta\right)\right)+h\,.

Note the following:

•

Since, for any other section $\tau^{\prime}=\mathrm{e}^{\mathrm{i}\mathring{\phi}}\tau$ , the corresponding change of coordinate is $\phi^{\prime}=\phi-\mathring{\phi}$ , the expression (4.1) does not depend on the choice of trivialisation $\sigma$ .
•

Under a change of contact form $\widehat{\theta}=\mathrm{e}^{\varphi}\theta$ for smooth function $\varphi$ on $\mathcal{M}$ , the metric transforms conformally as $\widetilde{g}_{\widehat{\theta}}^{(\alpha)}=\mathrm{e}^{\varphi}\widetilde{g}_{\theta}^{(\alpha)}$ .

We therefore have constructed a conformal class $\widetilde{\mathbf{c}}^{(\alpha)}$ of Lorentzian metrics in $\widetilde{\mathcal{M}}$ that includes $\alpha$ -Fefferman metrics. In addition, $\widetilde{\mathbf{c}}^{(\alpha)}$ admits a canonical conformal Killing field $\widetilde{k}$ , i.e. $\mathsterling_{\widetilde{k}}\widetilde{\bm{g}}=0$ , namely the generator $\widetilde{k}=\frac{\partial}{\partial\phi}$ of the fibers of $\widetilde{\mathcal{M}}\rightarrow\mathcal{M}$ . In fact, it is Killing for each of the Fefferman metrics in $\widetilde{\mathbf{c}}^{(\alpha)}$ , i.e. $\mathsterling_{\widetilde{k}}\widetilde{g}_{\theta}^{(\alpha)}=0$ for any contact form $\theta$ for $(H,J)$ .

We shall refer to $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}}^{(\alpha)},\widetilde{k})\longrightarrow(\mathcal{M},H,J)$ as the $\alpha$ -Fefferman space of $(\mathcal{M},H,J)$ .

When $\alpha=0$ , the conformal structure $\widetilde{\mathbf{c}}^{(0)}$ is identical to the one introduced by [Lei07].

When $\|\mathsf{N}\|^{2}=0$ , in which case $(H,J)$ is involutive, the parameter $\alpha$ becomes irrelevant, and we will simply talk of Fefferman metrics or conformal structure.

These definitions can also be generalised to conformal structure of signature $(p+1,q+1)$ with $p,q$ even, but if $pq\neq 0$ , $\|\mathsf{N}\|^{2}=0$ no longer implies the involutivity of $(H,J)$ .

We begin with Sparling’s characterisation of Fefferman spaces for CR manifolds.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}})$ be an oriented and time-oriented Lorentzian conformal manifold of dimension $n+2=2m+2$ . Then $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}})$ is locally conformally isometric to the Fefferman space of a CR manifold if and only if it admits a null conformal Killing field $\widetilde{k}$ and the following integrability conditions are satisfied:

\displaystyle\frac{1}{n^{2}}(\widetilde{\nabla}_{a}\widetilde{k}^{a})^{2}-\widetilde{\mathsf{P}}_{ab}\widetilde{k}^{a}\widetilde{k}^{b}-\frac{1}{n}\widetilde{k}^{a}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\widetilde{k}^{b}<0\,,

\displaystyle\widetilde{k}^{a}\widetilde{\mathsf{W}}_{abcd}=0\,,

\displaystyle\widetilde{k}^{a}\widetilde{\mathsf{Y}}_{abc}=0\,,

where $\widetilde{\nabla}$ is the Levi–Civita connection of any metric in $\widetilde{\mathbf{c}}$ with Schouten tensor $\widetilde{\mathsf{P}}_{ab}$ , Cotton tensor $\widetilde{\mathsf{Y}}_{abc}$ and Weyl tensor $\widetilde{\mathsf{W}}_{abcd}$ .

We next state characterisations of $\alpha$ -Fefferman spaces for almost CR manifolds. The proof has been relegated to Appendix B.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}})$ be an oriented and time-oriented Lorentzian conformal manifold of dimension $n+2=2m+2$ . Then, for any $\alpha\in\mathbf{R}$ , $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}})$ is locally conformally isometric to an $\alpha$ -Fefferman space of an almost CR structure if and only if it admits a null conformal Killing field $\widetilde{k}$ and the following integrability conditions are satisfied:


	$\displaystyle\frac{1}{n^{2}}(\widetilde{\nabla}_{a}\widetilde{k}^{a})^{2}-\widetilde{\mathsf{P}}_{ab}\widetilde{k}^{a}\widetilde{k}^{b}-\frac{1}{n}\widetilde{k}^{a}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\widetilde{k}^{b}<0\,,$		(4.2a)
	$\displaystyle\widetilde{k}^{a}\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{d}=\frac{1-\alpha}{8(2m+1)}\bm{\widetilde{\kappa}}_{b}\bm{\widetilde{\kappa}}_{c}\\|\widetilde{\mathsf{W}}(\widetilde{k})\\|^{2}\,,$		(4.2b)
	$\displaystyle\widetilde{k}^{a}\widetilde{\mathsf{Y}}_{abc}\widetilde{k}^{c}=\frac{1-\alpha}{16(2m+1)}\bm{\widetilde{\kappa}}_{b}\widetilde{k}^{c}\widetilde{\nabla}_{c}\\|\widetilde{\mathsf{W}}(\widetilde{k})\\|^{2}\,,$		(4.2c)
	$\displaystyle\widetilde{\mathsf{W}}_{ab}{}^{cd}\widetilde{\bm{\tau}}_{cd}-2\widetilde{k}^{c}\widetilde{\mathsf{Y}}_{cab}-\frac{1}{2}\left(\widetilde{\bm{\tau}}_{c[a}\widetilde{k}^{d}\widetilde{\mathsf{W}}_{b]d}{}^{ef}\widetilde{\mathsf{W}}_{efg}{}^{c}\widetilde{k}^{g}-\widetilde{\bm{\kappa}}_{[a}\widetilde{k}^{c}\widetilde{\mathsf{W}}_{b]c}{}^{de}\widetilde{\mathsf{Y}}_{fde}\widetilde{k}^{f}\right)$
	$\displaystyle\qquad\qquad\qquad=\frac{1}{4(2m+1)}\left(1-\frac{m^{2}-1}{m(m+2)}\alpha\right)\widetilde{\nabla}_{[a}\left(\widetilde{\bm{\kappa}}_{b]}\\|\widetilde{\mathsf{W}}(\widetilde{k})\\|^{2}\right)\,,$		(4.2d)

where $\widetilde{\nabla}$ is the Levi–Civita connection of any choice of metric in $\widetilde{\mathbf{c}}$ with Schouten tensor $\widetilde{\mathsf{P}}_{ab}$ , Cotton tensor $\widetilde{\mathsf{Y}}_{abc}$ and Weyl tensor $\widetilde{\mathsf{W}}_{abcd}$ ,

\displaystyle\|\widetilde{\mathsf{W}}(\widetilde{k})\|^{2}

\displaystyle:=\widetilde{k}^{a}\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{e}\widetilde{\mathsf{W}}_{e}{}^{bcd}\,,

and $\bm{\widetilde{\kappa}}_{a}=\widetilde{\bm{g}}_{ab}\widetilde{k}^{b}$ , $\widetilde{\bm{\tau}}_{ab}=\widetilde{\nabla}_{[a}\widetilde{\bm{\kappa}}_{b]}$ .

Conditions (4.2a), (4.2b), (4.2c) and (4.2d) are all conformally invariant, as can be seen from the transformations (3.3) and $\widehat{\widetilde{\bm{\tau}}}_{ab}=\widetilde{\bm{\tau}}_{ab}+2\,\widetilde{\Upsilon}_{[a}\widetilde{\bm{\kappa}}_{b]}$ .

In the involutive case, one also has a distinguished spinor field that satisfies the so-called twistor equation [Lew91, Bau, ČG08]. This spinor field is pure, i.e. it annihilates a totally null complex $(m+1)$ -plane distribution, which, in the present context, is none other than the Robinson structure arising from the CR structure. In the non-involutive case, there is no distinguished twistor-spinor field associated to $\widetilde{k}$ . If there were one, the integrability condition for such a twistor-spinor would imply $\widetilde{k}^{a}\widetilde{\mathsf{W}}_{abcd}=0$ , which would contradict the non-involutivity of $(\mathcal{M},H,J)$ by Theorem 4.5.

Henceforth, and for later convenience, which will become apparent in the subsequent sections, we refer to a $1$ -Fefferman space simply as a Fefferman space, i.e. a Fefferman metric will be denoted $\widetilde{g}_{\theta}$ instead of $\widetilde{g}_{\theta}^{(1)}$ , and the Fefferman conformal structure $\widetilde{\mathbf{c}}$ instead of $\widetilde{\mathbf{c}}^{(1)}$ .

We now generalise the Fefferman conformal structure as follows.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{k})\longrightarrow(\mathcal{M},H,J)$ be a Fefferman space. Let $\widetilde{\xi}$ be a semi-basic one-form on $\widetilde{\mathcal{M}}$ , that is, $\widetilde{k}\makebox[7.0pt]{\rule{6.0pt}{0.3pt}\rule{0.3pt}{5.0pt}}\,\widetilde{\xi}=0$ . Given a Fefferman metric $\widetilde{g}_{\theta}$ in $\widetilde{\mathbf{c}}$ , we define the Fefferman metric perturbed by $\widetilde{\xi}$ as

\displaystyle\widetilde{g}_{\theta,\widetilde{\xi}}

\displaystyle=\widetilde{g}_{\theta}+4\theta\odot\widetilde{\xi}\,.

This naturally extends to a conformal structure $\widetilde{\mathbf{c}}_{\widetilde{\xi}}$ , which we refer to as the Fefferman conformal structure perturbed by $\widetilde{\xi}$ , and we call $\widetilde{\xi}$ the perturbation one-form of $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{k})$ , and the triple $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}}_{\widetilde{\xi}},\widetilde{k})$ as a perturbed Fefferman space.

Since $\widetilde{\xi}$ lives on a circle bundle, its components can be Fourier expanded, and in fact, the Fourier coefficients should be understood as trivialised CR densities as the lemma below makes clear. Example 5.4 should clarify any ambiguity in the notation adopted.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}}_{\widetilde{\xi}},\widetilde{k})\longrightarrow(\mathcal{M},H,J)$ be a perturbed Fefferman space. For any subsets $\mathcal{I}\subset\mathbf{Z}$ , $\mathcal{J}\subset\mathbf{Z}_{\geq 0}$ , consider the tuple $\left(\bm{\xi}^{(i)}_{\alpha},[\nabla,\bm{\xi}^{(j)}_{0}]\right)_{i\in\mathcal{I},j\in\mathcal{J}}$ where

(1)

for $k\in\mathcal{I}$ , $\bm{\xi}_{\alpha}^{(k)}\in\Gamma(\mathcal{E}_{\alpha}\left(\tfrac{k}{2},-\tfrac{k}{2}\right))$ ,

(2)

for $k\in\mathcal{J}$ , $[\nabla,\bm{\xi}^{(k)}_{0}]$ denotes the equivalence class $(\nabla,\bm{\xi}_{0}^{(k)})\sim(\widehat{\nabla},\widehat{\bm{\xi}}_{0}^{(k)})$ where $\nabla,\widehat{\nabla}$ are Webster connections related by $\widehat{\nabla}=\nabla+\Upsilon$ for some exact $\Upsilon$ , and for $k\in\mathcal{I}$ , $\bm{\xi}_{0}^{(k)}$ and $\widehat{\bm{\xi}}_{0}^{(k)}$ are sections of $\mathcal{E}\left(\tfrac{k}{2}-1,-\tfrac{k}{2}-1\right)$ related by

\displaystyle\widehat{\bm{\xi}}{}^{(k)}_{0}

\displaystyle=\bm{\xi}^{(k)}_{0}-\mathrm{i}\bm{\xi}^{(k)}_{\alpha}\Upsilon^{\alpha}+\mathrm{i}\bm{\xi}^{(k)}_{\bar{\alpha}}\Upsilon^{\bar{\alpha}}\,,

(5.1)

with the understanding that, for $i\in\mathcal{I},j\in\mathcal{J}$ , $\bm{\xi}_{\bar{\alpha}}^{(i)}=\overline{\bm{\xi}_{\alpha}^{(-i)}}$ , $\bm{\xi}_{0}^{(j)}=\overline{\bm{\xi}_{0}^{(-j)}}$ . In particular $\bm{\xi}_{0}^{(0)}$ is real.

Choose some non-vanishing $\sigma\in\Gamma(\mathcal{E}(1,0))$ to trivialise $\widetilde{\mathcal{M}}\rightarrow\mathcal{M}$ with fibre coordinate $\phi$ , and set

\displaystyle\xi_{\alpha}^{(k)}

\displaystyle=\bm{\xi}_{\alpha}^{(k)}\sigma^{-\tfrac{k}{2}}\overline{\sigma}^{\tfrac{k}{2}}\,,

\displaystyle\xi_{\bar{\alpha}}^{(k)}

\displaystyle=\bm{\xi}_{\bar{\alpha}}^{(k)}\overline{\sigma}^{-\tfrac{k}{2}}\sigma^{\tfrac{k}{2}}\,,

\displaystyle\xi_{0}^{(k)}

\displaystyle=\bm{\xi}_{0}^{(k)}\sigma^{1-\tfrac{k}{2}}\overline{\sigma}^{1+\tfrac{k}{2}}\,.

(5.2)

Then, viewing

\displaystyle\widetilde{\xi}_{\alpha}

\displaystyle=\sum_{k\in\mathcal{I}}\xi_{\alpha}^{(k)}\mathrm{e}^{k\mathrm{i}\phi}\,,

\displaystyle\widetilde{\xi}_{\bar{\alpha}}

\displaystyle=\sum_{k\in\mathcal{I}}\xi_{\bar{\alpha}}^{(-k)}\mathrm{e}^{-k\mathrm{i}\phi}\,,

\displaystyle\widetilde{\xi}_{0}

\displaystyle=\sum_{k\in-\mathcal{J}\cup\mathcal{J}}\xi_{0}^{(k)}\mathrm{e}^{k\mathrm{i}\phi}\,,

(5.3)

as components with respect to some adapted coframe $(\theta,\theta^{\alpha},\overline{\theta}{}^{\bar{\alpha}})$ with $\theta=(\sigma\overline{\sigma})^{-1}\bm{\theta}$ , the semi-basic one-form

\displaystyle\widetilde{\xi}

\displaystyle=\widetilde{\xi}_{\alpha}\theta^{\alpha}+\widetilde{\xi}_{\bar{\alpha}}\overline{\theta}{}^{\bar{\alpha}}+\widetilde{\xi}_{0}\theta\,,

(5.4)

is well-defined on $\widetilde{\mathcal{M}}$ .

Conversely, any semi-basic one-form on $\widetilde{\mathcal{M}}$ arises in this way.

Starting from the tuple $\left(\bm{\xi}^{(i)}_{\alpha},[\nabla,\bm{\xi}^{(j)}_{0}]\right)_{i\in\mathcal{I},j\in\mathcal{J}}$ , we only need to check that (5.4) and (5.3) are well-defined. First, $\widetilde{\xi}$ is real by virtue of from the reality conditions on $\left(\bm{\xi}^{(i)}_{\alpha},[\nabla,\bm{\xi}^{(j)}_{0}]\right)_{i\in\mathcal{I},j\in\mathcal{J}}$ . Next, $\widetilde{\xi}$ does not depend on the choice of trivialisation, since under the transformation $\sigma^{\prime}=\mathrm{e}^{-\mathrm{i}\mathring{\phi}}\sigma$ , $\phi^{\prime}=\phi-\mathring{\phi}$ , the relations (5.1) tell us that the Fourier coefficients (5.2) transform as

\displaystyle{\xi^{\prime}}_{\alpha}^{(k)}

\displaystyle=\mathrm{e}^{k\mathrm{i}\mathring{\phi}}\xi_{\alpha}^{(k)}\,,

\displaystyle{\xi^{\prime}}_{0}^{(k)}

\displaystyle=\mathrm{e}^{k\mathrm{i}\mathring{\phi}}\xi_{0}^{(k)}\,.

Finally, $\widetilde{\xi}$ does not depend on the choice of adapted coframe as follows from (5.1). The converse works analogously. ∎

We shall refer to the tuple $\left(\bm{\xi}^{(i)}_{\alpha},[\nabla,\bm{\xi}^{(j)}_{0}]\right)_{i\in\mathcal{I},j\in\mathcal{J}}$ given in Lemma 5.2, where $\mathcal{I}\subset\mathbf{Z}$ , $\mathcal{J}\subset\mathbf{Z}_{\geq 0}$ , as the CR data associated to the perturbation one-form $\widetilde{\xi}$ .

A perturbation one-form $\widetilde{\xi}$ with CR data $\left(\bm{\xi}_{\alpha}^{(0)},[\nabla,\bm{\xi}_{0}^{(0)}],\bm{\xi}_{0}^{(2k)}\right)_{k=1,\ldots,m+1}$ means that with a choice of trivialisation $\sigma$ of $\widetilde{\mathcal{M}}$ with fibre coordinate $\phi$ , and adapted coframe $(\theta,\theta^{\alpha},\overline{\theta}{}^{\bar{\alpha}})$ with $\theta=(\sigma\overline{\sigma})^{-1}\bm{\theta}$ , the one-form $\widetilde{\xi}$ is given by

\displaystyle\widetilde{\xi}=\xi_{\alpha}^{(0)}\theta^{\alpha}+\xi_{\bar{\alpha}}^{(0)}\overline{\theta}{}^{\bar{\alpha}}+\sum_{k=-m-1}^{m+1}\xi_{0}^{(2k)}\mathrm{e}^{2k\mathrm{i}\phi}\theta\,,

where the coefficients are related to the CR data via (5.2). Note that, for $k\neq 0$ , $\widehat{\bm{\xi}}_{0}^{(2k)}={\bm{\xi}}_{0}^{(2k)}$ under a change of contact forms, which justifies our notation for the tuple.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{k})\rightarrow(\mathcal{M},H,J)$ be a Fefferman space. It is easy to check that two perturbations $\widetilde{\mathbf{c}}_{\widetilde{\xi}}$ and $\widetilde{\mathbf{c}}_{\widetilde{\xi}^{\prime}}$ are locally conformally isometric if and only if their corresponding CR data $\left(\bm{\xi}^{(i)}_{\alpha},[\nabla,\bm{\xi}^{(j)}_{0}]\right)_{i\in\mathcal{I},j\in\mathcal{J}}$ and $\left(\bm{\xi}^{\prime}{}^{(i)}_{\alpha},[\nabla,\bm{\xi}^{\prime}{}^{(j)}_{0}]\right)_{i\in\mathcal{I}^{\prime},j\in\mathcal{J}^{\prime}}$ are such that $\mathcal{I}^{\prime}=\mathcal{I}$ and $\mathcal{J}^{\prime}=\mathcal{J}$ , and

	$\displaystyle\bm{\xi}^{\prime}{}^{(0)}_{\alpha}$	$\displaystyle=\bm{\xi}^{(0)}_{\alpha}+\nabla_{\alpha}\mathring{\phi}\,,$	$\displaystyle[\nabla,\bm{\xi}^{\prime}{}^{(0)}_{0}]$	$\displaystyle=[\nabla,\bm{\xi}^{(0)}_{0}+\nabla_{0}\mathring{\phi}]\,,$
	$\displaystyle\bm{\xi}^{\prime}{}^{(k)}_{\alpha}$	$\displaystyle=\mathrm{e}^{k\mathrm{i}\mathring{\phi}}\bm{\xi}^{(k)}_{\alpha}\,,$	$\displaystyle[\nabla,\bm{\xi}^{\prime}{}^{(k)}_{0}]$	$\displaystyle=[\nabla,\mathrm{e}^{k\mathrm{i}\mathring{\phi}}\bm{\xi}^{(k)}_{0}]\,,$	$\displaystyle k\neq 0\,,$

for some smooth function $\mathring{\phi}$ on $(\mathcal{M},H,J)$ . In the special case where $\widetilde{\xi}^{\prime}=0$ , we conclude that $\widetilde{\mathbf{c}}_{\widetilde{\xi}}$ and $\widetilde{\mathbf{c}}$ are locally conformally isometric if and only if $\widetilde{\xi}$ is closed (i.e. locally exact), and one can use the criterion of Lemma 2.2 on $\xi_{\alpha}=\xi_{\alpha}^{(0)}$ and $\xi_{0}=\xi_{0}^{(0)}$ to verify this property. See also [Lei10]. One can extend these arguments as done in [Gra87] to verify that an obstruction to global conformal isometry between two perturbed Fefferman spaces is the cohomology class $H^{1}(\mathcal{M},S^{1})$ .

Finally, we state the next self-evident result without proof.

Any perturbed Fefferman space is a twist-induced nearly Robinson geometry with non-shearing congruence.

In the next section, we investigate under which conditions the converse is true.

Consider a $(2m+2)$ -dimensional twist-induced nearly Robinson geometry $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})\longrightarrow(\mathcal{M},H,J)$ with non-shearing congruence $\widetilde{\mathcal{K}}$ . We refer the reader to Sections 3.1 and 3.2 for the general setup and notation concerning these geometries, and more particularly Section 3.3. Since a perturbed Fefferman metric is a particular case of such an optical geometry, it will be convenient to write

\displaystyle\widetilde{g}_{\theta,\widetilde{\xi}}

\displaystyle=4\theta\odot\widetilde{\lambda}+h\,,

\displaystyle\widetilde{\lambda}

\displaystyle:=\widetilde{\omega}_{\theta}-\left(\frac{1}{m+2}\mathsf{P}+\frac{1}{2m(m+1)}\|\mathsf{N}\|^{2}\right)\theta+\widetilde{\xi}\,,

(5.5)

where $\widetilde{\omega}_{\theta}$ is the induced Webster connection one-form on $\widetilde{\mathcal{M}}$ with Webster–Schouten tensor $\mathsf{P}$ and Nijenhuis tensor $\mathsf{N}_{\alpha\beta\gamma}$ . Just as we did in Lemma 5.2, we can Fourier expand the components of $\widetilde{\lambda}$ with respect to an adapted coframe.

We start with a technical lemma.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})\accentset{\varpi}{\longrightarrow}(\mathcal{M},H,J)$ be a twist-induced nearly Robinson geometry with twisting non-shearing congruence. Let $\widetilde{g}_{\theta}\in\accentset{n.e.}{\widetilde{\mathbf{c}}}$ for some pseudo-hermitian structure $\theta$ so that $\widetilde{g}_{\theta}$ is given by (3.8). Suppose that

\displaystyle\widetilde{\lambda}_{\alpha}

\displaystyle=\sum_{k\in\mathcal{I}}\lambda_{\alpha}^{(k)}\mathrm{e}^{k\mathrm{i}\phi}\,,

\displaystyle\widetilde{\lambda}_{\bar{\alpha}}

\displaystyle=\sum_{k\in\mathcal{I}}\lambda_{\bar{\alpha}}^{(-k)}\mathrm{e}^{-k\mathrm{i}\phi}\,,

\displaystyle\widetilde{\lambda}_{0}

\displaystyle=\sum_{k\in-\mathcal{J}\cup\mathcal{J}}\lambda_{0}^{(k)}\mathrm{e}^{k\mathrm{i}\phi}\,,

(5.6)

for some $\mathcal{I}\subset\mathbf{Z}$ , $\mathcal{J}\subset\mathbf{Z}_{\geq 0}$ . Then $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})$ is locally conformally isometric to a perturbed Fefferman space $(\widetilde{\mathcal{M}}^{\prime},\widetilde{\mathbf{c}}^{\prime}_{\widetilde{\xi}},\widetilde{k}^{\prime})\longrightarrow(\mathcal{M},H,J)$ with perturbation one-form $\widetilde{\xi}$ determined by the CR data $\left(\bm{\xi}_{\alpha}^{(i)},[\nabla,\bm{\xi}_{0}^{(j)}]\right)_{i\in\mathcal{I},j\in\mathcal{J}}$ , where, for each choice $\sigma\in\Gamma(\mathcal{E}(1,0))$ trivialising $\widetilde{\mathcal{M}}$ , we have


$\displaystyle\lambda_{\alpha}^{(0)}$	$\displaystyle=\mathrm{i}\sigma^{-1}\nabla_{\alpha}\sigma+\xi_{\alpha}^{(0)}\,,$	$\displaystyle\lambda_{0}^{(0)}$	$\displaystyle=\mathrm{i}\sigma^{-1}\nabla_{0}\sigma+\xi_{0}^{(0)}-\left(\frac{1}{m+2}\mathsf{P}+\frac{1}{2m(m+1)}\\|\mathsf{N}\\|^{2}\right)\,,$		(5.7a)
$\displaystyle\lambda_{\alpha}^{(k)}$	$\displaystyle=\xi_{\alpha}^{(k)}\,,$	$\displaystyle\lambda_{0}^{(k)}$	$\displaystyle=\xi_{0}^{(k)}\,,$	$\displaystyle k\neq 0\,.$	(5.7b)

where $\xi_{\alpha}^{(i)}$ and $\xi_{0}^{(j)}$ are given by (5.2).

Since $(\mathcal{M},H,J)$ is an almost CR manifold, we associate to it its conformal Fefferman space $(\widetilde{\mathcal{M}}^{\prime},\widetilde{\mathbf{c}}^{\prime},\widetilde{k}^{\prime})\accentset{\varpi^{\prime}}{\longrightarrow}(\mathcal{M},H,J)$ . To the pseudo-hermitian structure $\theta$ , we have an associated Webster connection $\nabla$ with induced connection one-form $\widetilde{\omega}_{\theta}$ on $\widetilde{\mathcal{M}}^{\prime}$ , and a corresponding Fefferman metric $\widetilde{g}^{\prime}_{\theta}$ in $\widetilde{\mathbf{c}}^{\prime}$ .

Next, choose a non-vanishing density $\sigma$ of weight $(1,0)$ such that $\theta=(\sigma\overline{\sigma})^{-1}\bm{\theta}$ and denote by $\phi^{\prime}$ the corresponding fibre coordinate on $\widetilde{\mathcal{M}}^{\prime}\rightarrow\mathcal{M}$ . The map that sends a point $\widetilde{p}$ in $\widetilde{\mathcal{M}}$ to $(\varpi(\widetilde{p}),\mathrm{e}^{\mathrm{i}\phi^{\prime}\circ\varpi(\widetilde{p})}\sigma^{-1})$ in $\widetilde{\mathcal{M}}^{\prime}$ is a bundle map $\iota$ from $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{k})$ to $(\widetilde{\mathcal{M}}^{\prime},\widetilde{\mathbf{c}}^{\prime},\widetilde{k}^{\prime})$ : clearly $\varpi^{\prime}\circ\iota=\varpi$ , and any change of affine parametrisation of $\widetilde{k}$ of the form $\widehat{\phi}=\phi-\mathring{\phi}$ induces a change of trivialisation as $\widehat{\sigma}=\mathrm{e}^{-\mathrm{i}\mathring{\phi}}\sigma$ , so $\iota$ does not depend on the choice of trivialisation.

Now define

	$\displaystyle\xi_{\alpha}^{(0)}$	$\displaystyle=\lambda_{\alpha}^{(0)}-\mathrm{i}\sigma^{-1}\nabla_{\alpha}\sigma\,,$	$\displaystyle\xi_{0}^{(0)}$	$\displaystyle=\lambda_{0}^{(0)}-\mathrm{i}\sigma^{-1}\nabla_{0}\sigma+\left(\frac{1}{m+2}\mathsf{P}+\frac{1}{2m(m+1)}\\|\mathsf{N}\\|^{2}\right)\,,$
	$\displaystyle\xi_{\alpha}^{(k)}$	$\displaystyle=\lambda_{\alpha}^{(k)}\,,$	$\displaystyle\xi_{0}^{(k)}$	$\displaystyle=\lambda_{0}^{(k)}\,,$	$\displaystyle k\neq 0\,.$

It is then straightforward to check that under an affine reparametrisation of the geodesics of $\widetilde{k}$ and a change of contact form, the induced transformations of $\xi_{0}^{(k)}$ and $\xi_{0}^{(k)}$ allow us to view them as the trivialisations of some CR data $\left(\bm{\xi}_{\alpha}^{(i)},[\nabla,\bm{\xi}_{0}^{(j)}]\right)_{i\in\mathcal{I},j\in\mathcal{J}}$ . We can therefore define a perturbation one-form $\widetilde{\xi}$ from this CR data, and an associated perturbed Fefferman metric

\displaystyle\widetilde{g}^{\prime}_{\theta,\widetilde{\xi}}

\displaystyle:=\widetilde{g}^{\prime}_{\theta}+4\theta\odot\widetilde{\xi}\,.

One can also verify that under a change of contact form, the metrics thus constructed rescale by the same conformal factor. Checking that $\iota$ is a conformal isometry that sends each metric $\widetilde{g}_{\theta}$ in $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ to a perturbed Fefferman metric $\widetilde{g}^{\prime}_{\theta,\widetilde{\xi}}$ , is routine. In particular, $\iota_{*}\widetilde{k}|_{\widetilde{p}}=\widetilde{k}^{\prime}|_{\iota(\widetilde{p})}$ at every point $\widetilde{p}$ of $\widetilde{\mathcal{M}}$ . ∎

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ be an optical geometry of dimension $2m+2>4$ with twisting non-shearing congruence of null geodesics. Then, we already know, by Theorem 3.1, that it induces a nearly Robinson structure $(\widetilde{N},\widetilde{k})$ where $\widetilde{k}$ is the distinguished section of $\widetilde{K}$ of Section 3.3 if and only the Weyl tensor satisfies

\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}\widetilde{v}^{d}

\displaystyle=0\,,

for any

\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})

(5.8)

We presently examine further conformally invariant degeneracy curvature conditions.

	$\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}$	$\displaystyle=0\,,$	for any $\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})$ ,		(5.9)
	$\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{\nabla}^{c}\widetilde{k}^{d}$	$\displaystyle=0\,,$	for any $\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})$ ,		(5.10)

if and only if the twist of $\widetilde{\mathcal{K}}$ induces a nearly Robinson structure, and

\displaystyle\mathsterling_{\widetilde{k}}\widetilde{g}(\widetilde{v},\cdot)

\displaystyle\propto\widetilde{g}(\widetilde{v},\cdot)\,,

for any

\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})

(5.11)

Condition (5.9) implies (5.8), which tells us that we can view our conformal manifold as a twist-induced nearly Robinson geometry $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})$ as in Section 3.3. We work with a metric $\widetilde{g}_{\theta}$ in $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ for some pseudo-hermitian structure $\theta$ as given by (3.8). It is shown in [TC22] that the Weyl tensor satisfies (5.9) if and only the component $\widetilde{\lambda}_{\alpha}$ satisfies

\displaystyle\widetilde{\lambda}_{\alpha}

\displaystyle=\begin{cases}\lambda_{\alpha}+\mu_{\alpha}\phi\,,&m=2\,,\\ \lambda_{\alpha}+\mu_{\alpha}\mathrm{e}^{\frac{2m-4}{2m-1}\mathrm{i}\phi}\,,&m>2\,,\end{cases}

(5.12)

for some $\lambda_{\alpha},\mu_{\alpha}\in\Gamma(\mathcal{E}_{\alpha})$ . On the other hand, using Appendix A of [TC22], we can compute, in the obvious notation,

\displaystyle\widetilde{\mathsf{W}}_{\beta}{}^{\beta 0}{}_{\alpha}

\displaystyle=-\frac{1}{4m}\left(\ddot{\widetilde{\lambda}}_{\alpha}+2(2m^{2}+m-2)\mathrm{i}\dot{\widetilde{\lambda}}_{\alpha}\right)\,,

and so, condition (5.10) is equivalent to

\displaystyle\widetilde{\lambda}_{\alpha}

\displaystyle=\lambda_{\alpha}+\nu_{\alpha}\mathrm{e}^{-2(2m^{2}+m-2)\mathrm{i}\phi}\,,

(5.13)

for some $\nu_{\alpha}\in\Gamma(\mathcal{E}_{\alpha})$ . By (5.12) and (5.13), it immediately follows that $\widetilde{\lambda}_{\alpha}=\lambda_{\alpha}$ . But, for any $\widetilde{v}\in\Gamma(\langle\widetilde{k}\rangle^{\perp})$ , we have that

\displaystyle\mathsterling_{\widetilde{k}}\widetilde{g}_{\theta}(\widetilde{v},\cdot)

\displaystyle=4\theta\odot\left(\widetilde{k}\makebox[7.0pt]{\rule{6.0pt}{0.3pt}\rule{0.3pt}{5.0pt}}\,\mathrm{d}\widetilde{\lambda}+\mathrm{d}(\widetilde{k}\makebox[7.0pt]{\rule{6.0pt}{0.3pt}\rule{0.3pt}{5.0pt}}\,\widetilde{\lambda})\right)(\widetilde{v},\cdot)=2\dot{\widetilde{\lambda}}_{\alpha}\theta^{\alpha}(\widetilde{v})\theta=0\,,

which establishes condition (5.11). The converse works in the same way. ∎

Note that, for any $\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})$ , we have

\displaystyle(n-1)\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}\widetilde{\mathsf{Y}}_{abc}

\displaystyle=-\widetilde{\nabla}^{c}(\widetilde{k}^{d}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{\mathsf{W}}_{cdab})-(\widetilde{\nabla}^{d}\widetilde{v}^{b})\widetilde{k}^{c}\widetilde{\mathsf{W}}_{cdab}\widetilde{k}^{a}+\frac{3}{2}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{\mathsf{W}}_{abcd}\widetilde{\nabla}^{c}\widetilde{k}^{d}\,.

Assume (5.9), which can also be re-expressed as $\widetilde{k}^{c}\widetilde{\mathsf{W}}_{cdab}\widetilde{k}^{a}=\widetilde{f}\widetilde{\kappa}_{d}\widetilde{\kappa}_{b}$ for some function $\widetilde{f}$ . Then the above equation reduces to $(n-1)\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}\widetilde{\mathsf{Y}}_{abc}=\frac{3}{2}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{\mathsf{W}}_{abcd}\widetilde{\nabla}^{c}\widetilde{k}^{d}$ . Hence, (5.10) is equivalent to

\displaystyle\widetilde{\mathsf{Y}}_{abc}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}

\displaystyle=0\,,

for any

\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})

which is conformally invariant provided that (5.9) holds.

In the case where $2m+2=4$ , conditions (5.9) and (5.10) turn out to be equivalent, and we find that $\widetilde{\lambda}=\lambda_{\alpha}+\mu_{\alpha}\mathrm{e}^{-2\mathrm{i}\phi}$ , which is weaker, i.e. the expression can readily be interpreted as a Fourier expansion of $\widetilde{\lambda}_{\alpha}$ viewed as a periodic function with period $\pi$ — this is treated in [TC23].

In the present context, the combined conditions (5.9) and (5.10) may be viewed as an even higher-dimensional generalisation of the notion of repeated principal null direction — see Section 3.3 of [TC23]. However, assuming merely (5.9), equation (5.12) tells us that $\widetilde{\lambda}_{\alpha}$ is clearly not periodic in $\phi$ in the case $m=2$ , while in the case $m>2$ , the period $\frac{2m-4}{2m-1}\pi$ is not an integer, which is problematic if one ultimately desires to interpret $\widetilde{\mathcal{M}}$ as Fefferman’s bundle over $\mathcal{M}$ . It is then necessary to impose the additional requirement (5.10) to force $\widetilde{\lambda}$ to be interpretable as an appropriate Fourier expansion in $\phi$ .

To fully integrate out the fibre dependence of $\widetilde{\lambda}$ , we need the following stronger condition.

	$\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{k}^{c}$	$\displaystyle=0\,,$	for any $\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})$ ,		(5.14)
	$\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{\nabla}^{c}\widetilde{k}^{d}$	$\displaystyle=0\,,$	for any $\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})$ .		(5.15)

Then $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ is locally conformally isometric to a perturbed Fefferman space $(\widetilde{\mathcal{M}}^{\prime},\widetilde{\mathbf{c}}^{\prime}_{\widetilde{\xi}},\widetilde{k}^{\prime})\longrightarrow(\mathcal{M},H,J)$ , and the perturbation one-form $\widetilde{\xi}$ is determined by the CR data $\left(\bm{\xi}_{\alpha}^{(0)},[\nabla,\bm{\xi}_{0}^{(0)}]\right)$ where

\displaystyle\bm{\xi}_{0}^{(0)}

\displaystyle=\frac{\mathrm{i}}{m}\left(\nabla_{\alpha}\bm{\xi}^{\alpha}_{(0)}-\nabla^{\alpha}\bm{\xi}_{\alpha}^{(0)}\right)\,,

(5.16)

In particular, $\widetilde{k}$ is conformal Killing.

Again, condition (5.9) implies (5.8), so that we view our conformal manifold as a twist-induced nearly Robinson geometry $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})$ as in Section 3.3. Let $\widetilde{g}_{\theta}$ be any metric in $\accentset{n.e.}{\widetilde{\mathbf{c}}}$ for some pseudo-hermitian structure $\theta$ as given by (3.8). Clearly, the pair of conditions (5.14) and (5.15) implies the hypotheses of Proposition 5.8, which allows us to write $\widetilde{\lambda}_{\alpha}=\lambda_{\alpha}^{(0)}$ for some $\lambda_{\alpha}^{(0)}\in\Gamma(\mathcal{E}_{\alpha})$ . To integrate out the $\phi$ -dependence of $\widetilde{\lambda}_{0}$ , we must make full use of (5.14) — incidentally, this constraint was already used in Theorem 4.6 — and compute $\widetilde{W}(\widetilde{k},\widetilde{\ell},\widetilde{k},\widetilde{\ell})$ , where $\widetilde{\ell}$ is such that $\widetilde{g}_{\theta}(\widetilde{k},\widetilde{\ell})=1$ . We find that the component $\widetilde{\lambda}_{0}$ is constant along $\widetilde{k}$ , and

\displaystyle\widetilde{\lambda}_{0}

\displaystyle=\lambda_{0}^{(0)}:=\frac{\mathrm{i}}{m}\left(\nabla_{\alpha}\lambda^{\alpha}_{(0)}-\nabla^{\alpha}\lambda_{\alpha}^{(0)}\right)+\frac{1}{2m(m+1)}\left(\mathsf{Sc}-\|\mathsf{N}\|^{2}\right)\,.

We can now use (5.7) and apply the commutation relation (2.4a) to conclude

\displaystyle\xi_{\alpha}^{(0)}

\displaystyle=\lambda_{\alpha}^{(0)}-\mathrm{i}\sigma^{-1}\nabla_{\alpha}\sigma\,,

\displaystyle\xi_{0}^{(0)}

\displaystyle=\frac{\mathrm{i}}{m}\left(\nabla_{\alpha}\xi^{\alpha}_{(0)}-\nabla^{\alpha}\xi_{\alpha}^{(0)}\right)\,.

Invoking Lemma 5.7 completes the proof. ∎

We note that if the perturbation one-form is (locally) exact then (5.16) holds by Lemma 2.2. It also means that the $\widetilde{\mathbf{c}}$ is locally conformally isometric to $\widetilde{\mathbf{c}}$ . In particular, the addition curvature condition (4.2d) holds with $\alpha=1$ .

In dimension four, (5.14) implies (5.15). We may therefore regard the pair of conditions (5.14) and (5.15) as being an even higher-dimensional analogue of the notion of $\widetilde{k}$ being a triple repeated null direction — see Section 3.3 of [TC23].

It is proved in [TC23] that in dimension four, full integration of the components of $\widetilde{\lambda}_{\alpha}$ and $\widetilde{\lambda}_{0}$ is possible under weaker curvature conditions, namely that the Weyl tensor and the Bach tensor satisfy

	$\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}$	$\displaystyle=0\,,$	for any $\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})$ ,
	$\displaystyle\widetilde{\mathsf{B}}_{ab}\widetilde{k}^{a}\widetilde{k}^{b}$	$\displaystyle=0\,,$	for any $\widetilde{k}\in\Gamma(\widetilde{K})$ ,

respectively. In higher dimensions, the Bach tensor is not conformally invariant any more, but there is a conformally invariant analogue, known as the Fefferman–Graham obstruction tensor as introduced in [FG85], see also [GP06], which we shall denote by $\widetilde{\mathsf{F}\mkern-4.0mu\mathsf{G}}$ . This tensor is an obstruction to the existence of an Einstein metric in the conformal class. Its explicit form is rather complicated, and is dimension-dependent. We shall content ourselves to conjecture the following generalisation of Theorem 5.7 of [TC23] to dimensions $2m+2>4$ . The form of the perturbation one-form given below is motivated by Theorems 6.12 and 6.14 of the next section.

\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}

\displaystyle=0\,,

\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{\nabla}^{c}\widetilde{k}^{d}

\displaystyle=0\,,

for any

\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})

and the Fefferman–Graham obstruction tensor

\displaystyle\widetilde{\mathsf{F}\mkern-4.0mu\mathsf{G}}_{ab}\widetilde{k}^{a}\widetilde{k}^{b}

\displaystyle=0\,,

for any

\widetilde{k}\in\Gamma(\widetilde{K})

\displaystyle\bm{\xi}_{0}^{(0)}

\displaystyle=\frac{\mathrm{i}}{m}\left(\nabla_{\alpha}\bm{\xi}^{\alpha}_{(0)}-\nabla^{\alpha}\bm{\xi}_{\alpha}^{(0)}\right)\,.

We now investigate the consequences of the existence of the distinguished almost Lorentzian scales that we introduced in Section 3.4. For the time being, the dimension is assumed to be $2m+2\geq 4$ . We first describe the zero set of such scales.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})\accentset{\varpi}{\longrightarrow}(\mathcal{M},H,J)$ be a $(2m+2)$ -dimensional twist-induced nearly Robinson geometry with non-shearing congruence. Then in the neighbourhood $\widetilde{\mathcal{U}}$ of any point, we can always find an almost Lorentzian scale $\widetilde{\sigma}\in\Gamma(\widetilde{\mathcal{E}}[1])$ that satisfies

\displaystyle\widetilde{k}{}^{a}\widetilde{k}{}^{b}\left(\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\widetilde{\sigma}+\widetilde{\mathsf{P}}_{ab}\widetilde{\sigma}\right)

\displaystyle=0\,,

(6.1)

and whose zero set $\widetilde{\mathcal{Z}}$ consists of the union of sections of $\varpi:\widetilde{\mathcal{U}}\rightarrow\mathcal{U}:=\varpi(\widetilde{\mathcal{U}})\subset\mathcal{M}$ parametrised by the integers $\mathbf{Z}$ .

More specifically, there exists an affine parameter $\phi$ along the geodesics of $\widetilde{k}$ , unique up to transformations $\phi\mapsto\phi+k\pi$ for any $k\in\mathbf{Z}$ such that

\displaystyle\widetilde{\mathcal{Z}}=\left\{\phi=\tfrac{2k+1}{2}\pi\,|\,k\in\mathbf{Z}\right\}\,,

(6.2)

and a unique contact form $\theta$ for $(H,J)$ so that the metric $\widetilde{g}=\widetilde{\sigma}{}^{-2}\widetilde{\bm{g}}$ takes the form

\displaystyle\widetilde{g}

\displaystyle=\sec^{2}\phi\cdot\widetilde{g}_{\theta}\,,

off

\widetilde{\mathcal{Z}}

(6.3)

where $\widetilde{g}_{\theta}\in\accentset{n.e.}{\widetilde{\bm{c}}}$ is the metric associated to $\theta$ , and off $\widetilde{\mathcal{Z}}$ , the Ricci tensor of $\widetilde{g}$ satisfies

\displaystyle\widetilde{\mathsf{Ric}}(\widetilde{k},\widetilde{k})

\displaystyle=0\,.

(6.4)

In particular, we can write, up to sign,

\displaystyle\widetilde{\sigma}

\displaystyle=\cos\phi\cdot\widetilde{\sigma}_{\theta}\,,

(6.5)

where $\widetilde{\sigma}_{\theta}$ is the Lorentzian scale of $\widetilde{g}_{\theta}$ .

Locally, any almost Lorentzian scale $\widetilde{\sigma}$ on $\widetilde{\mathcal{M}}$ such that the Ricci tensor of $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ satisfies (6.4) arises in this way.

The existence of a density $\widetilde{\sigma}$ that satisfies (6.1) off its zero set $\widetilde{\mathcal{Z}}$ is equivalent to the existence of a metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ that satisfies (6.4) off $\widetilde{\mathcal{Z}}$ . This follows from the same argument used in the proof of Proposition 3.10. So we first show that a metric $\widetilde{g}$ with the required Ricci prescription exists. To this end, we may work on some open set $\widetilde{\mathcal{U}}$ and assume that

\displaystyle\widetilde{g}

\displaystyle=\mathrm{e}^{\widetilde{\varphi}}\widetilde{g}_{\theta}\,,

for some smooth function $\widetilde{\varphi}$ on $\widetilde{\mathcal{M}}$ , and where $\widetilde{g}_{\theta}\in\accentset{n.e.}{\widetilde{\mathbf{c}}}$ for some contact form $\theta$ . The condition that the Ricci tensor of $\widetilde{g}$ satisfies (6.4) is equivalent to $\widetilde{\varphi}$ satisfying $\ddot{\widetilde{\varphi}}-\dot{\widetilde{\varphi}}^{2}=1$ [LN90, TC22]. On $\widetilde{\mathcal{U}}\setminus\widetilde{\mathcal{Z}}$ , this has a unique solution, up to sign,

\displaystyle\mathrm{e}^{\widetilde{\varphi}}

\displaystyle=\mathring{\varrho}\cdot\cos(\phi-\mathring{\phi})\,,

(6.6)

for some smooth functions $\mathring{\varrho}$ and $\mathring{\phi}$ on $\mathcal{M}$ , where $\mathring{\varrho}$ is nowhere vanishing. We can then rescale $\theta$ by $\mathring{\varrho}$ , and reparametrise the geodesics of $\widetilde{k}$ as in (3.9), and this allows us to define $\widetilde{\mathcal{Z}}$ by (6.2). Then, the zero set of $\widetilde{\sigma}$ is precisely $\widetilde{\mathcal{Z}}$ , and off $\widetilde{\mathcal{Z}}$ , its associated metric is given by (6.3) with Ricci tensor satisfying the required property (6.4). Since $\widetilde{g}_{\theta}=\widetilde{\sigma}_{\theta}^{-2}\widetilde{\bm{g}}$ for some $\widetilde{\sigma}_{\theta}\in\Gamma(\widetilde{\mathcal{E}}_{+}[1])$ is regular on $\widetilde{\mathcal{U}}$ , the almost Lorentzian scale corresponding to $\widetilde{g}$ can be expressed by (6.5). ∎

Condition (6.4) is in fact conformally invariant if we restrict ourselves to conformal changes of the metric that are induced from changes of contact forms on $(\mathcal{M},H,J)$ — this can be seen from the transformation law (3.2) for the Schouten tensor. This therefore places no condition on the existence of any particular (almost) Lorentzian scales.

Since the zero set of the density $\widetilde{\sigma}$ in Proposition 6.1 is never empty, we obtain as a corollary:

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})\accentset{\varpi}{\longrightarrow}(\mathcal{M},H,J)$ be a $(2m+2)$ -dimensional twist-induced nearly Robinson geometry with non-shearing congruence. Then there is no global metric in $\widetilde{\mathbf{c}}$ whose Ricci tensor satisfies $\widetilde{\mathsf{Ric}}(\widetilde{k},\widetilde{k})=0$ .

We can refine our findings by assuming that our optical geometry is in fact a perturbed Fefferman conformal structure, which means that we work on a circle, rather than line, bundle. Let us return to Proposition 6.1. As described in Section 4, the fibre coordinate $\phi$ , the contact form $\theta$ and the perturbed Fefferman metric $\widetilde{g}_{\theta,\widetilde{\xi}}$ are all determined by a choice of non-vanishing section $\sigma\in\Gamma(\mathcal{E}(1,0))$ . We now define $\widehat{\sigma}:=z\sigma$ where $\mathring{z}=\mathring{\varrho}\mathrm{e}^{-\mathrm{i}\mathring{\phi}}$ with $\mathring{\varrho}$ and $\mathring{\phi}$ being the functions of integrations in (6.6). This induces a change of contact form, and thus a change of perturbed Fefferman metric $\widetilde{g}_{\widehat{\theta},\widetilde{\xi}}$ , and a change of fibre coordinate. For notational convenience, we drop the hats, i.e. $\sigma$ is our new density $\widehat{\sigma}$ . Then $\widetilde{\sigma}$ is an almost Lorentzian scale given by $\widetilde{\sigma}=\cos\phi\cdot\widetilde{\sigma}_{\theta}$ , where $\widetilde{\sigma}_{\theta}$ is the Lorentzian scale for $\widetilde{g}_{\theta,\widetilde{\xi}}$ , and it is not difficult to see that it must satisfy

\displaystyle\widetilde{k}{}^{a}\widetilde{k}{}^{b}\left(\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\widetilde{\sigma}+\widetilde{\mathsf{P}}_{ab}\widetilde{\sigma}\right)

\displaystyle=0\,,

by virtue of (6.4). Finally, since the fibre coordinate $\phi$ lies in $[-\pi,\pi)$ , the zero set of $\widetilde{\sigma}$ is now simply the hypersurfaces $\phi=\pm\frac{\pi}{2}$ , which, by definition of the Fefferman bundle, can be identified as the cross-sections $[\pm\mathrm{i}\sigma^{-1}]:\mathcal{M}\rightarrow\widetilde{\mathcal{M}}$ .

We can now reformulate Proposition 6.1 as follows:

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}}_{\widetilde{\xi}},\widetilde{k})\accentset{\varpi}{\longrightarrow}(\mathcal{M},H,J)$ be a $(2m+2)$ -dimensional perturbed Fefferman space. Then any choice of non-vanishing density $\sigma$ of weight $(1,0)$ on $\mathcal{M}$ determines an almost Lorentzian scale $\widetilde{\sigma}\in\Gamma(\widetilde{\mathcal{E}}[1])$ on $\widetilde{\mathcal{M}}$ that satisfies

\displaystyle\widetilde{k}{}^{a}\widetilde{k}{}^{b}\left(\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\widetilde{\sigma}+\widetilde{\mathsf{P}}_{ab}\widetilde{\sigma}\right)

\displaystyle=0\,,

(6.7)

and whose zero set $\widetilde{\mathcal{Z}}$ of $\widetilde{\sigma}$ consists of the union $\mathcal{Z}_{+}\cup\mathcal{Z}_{-}$ of the sections $\mathcal{Z}_{\pm}=\{[\pm\mathrm{i}\sigma^{-1}]:\mathcal{M}\rightarrow\widetilde{\mathcal{M}}\}$ for some unique, up to sign, density $\sigma$ of weight $(1,0)$ on $\mathcal{M}$ .

In particular, the metric $\widetilde{g}=\widetilde{\sigma}{}^{-2}\widetilde{\bm{g}}$ takes the form

\displaystyle\widetilde{g}

\displaystyle=\sec^{2}\phi\cdot\widetilde{g}_{\theta,\widetilde{\xi}}\,,

(6.8)

where $\widetilde{g}_{\theta,\widetilde{\xi}}$ is the perturbed Fefferman metric associated to the contact form $\theta=(\sigma\overline{\sigma})^{-1}\bm{\theta}$ , and $\phi$ is the fibre coordinate determined by $\sigma$ . Off $\widetilde{\mathcal{Z}}$ , the Ricci tensor of $\widetilde{g}$ satisfies

\displaystyle\widetilde{\mathsf{Ric}}(\widetilde{k},\widetilde{k})

\displaystyle=0\,.

(6.9)

Conversely, any almost Lorentzian scale $\widetilde{\sigma}$ on $\widetilde{\mathcal{M}}$ satisfying (6.7) arises from a non-vanishing density $\sigma\in\Gamma(\mathcal{E}(1,0))$ , unique up to sign.

The sign ambiguity of the density $\sigma$ in Proposition 6.4 is harmless since the change $\sigma\mapsto-\sigma$ leaves the contact form $\theta$ unchanged, and merely interchanges the two connected components $\widetilde{\mathcal{Z}}_{+}$ and $\widetilde{\mathcal{Z}}_{+}$ of the zero set of $\widetilde{\sigma}$ .

While $\widetilde{g}_{\theta}$ in Proposition 6.1 is regular on $\widetilde{\mathcal{U}}$ , in general there is no guarantee that it is periodic, which prevents us to conclude that $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})$ is a circle bundle as for perturbed Fefferman spaces.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})\accentset{\varpi}{\longrightarrow}(\mathcal{M},H,J)$ be a $(2m+2)$ -dimensional twist-induced nearly Robinson geometry with non-shearing congruence. Then there exists an almost Lorentzian scale $\widetilde{\sigma}\in\Gamma(\widetilde{\mathcal{E}}[1])$ that satisfies

\displaystyle\widetilde{k}{}^{a}\widetilde{v}{}^{b}\left(\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\widetilde{\sigma}+\widetilde{\mathsf{P}}_{ab}\widetilde{\sigma}\right)

\displaystyle=0\,,

for all

\widetilde{v}^{a}\in\Gamma(\langle\widetilde{k}\rangle^{\perp})

(6.10)

if and only if, off the zero set $\widetilde{\mathcal{Z}}$ of $\widetilde{\sigma}$ , the Ricci tensor of the metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ satisfies $\widetilde{\mathsf{Ric}}_{ab}\widetilde{k}^{a}\widetilde{v}^{b}=0$ . This in turn implies that the Cotton tensor satisfies $\widetilde{\mathsf{Y}}_{abc}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}=0$ for any $\widetilde{v}^{a}\in\Gamma(\langle\widetilde{k}\rangle^{\perp})$ .

That $\widetilde{\sigma}$ satisfying (6.10) defines the metric $\widetilde{g}$ off $\widetilde{\mathcal{Z}}$ with the stated degeneracy condition on its Ricci tensor is clear — see the proof of Proposition 3.10. Note that for any $\widetilde{v}\in\Gamma(\langle\widetilde{k}\rangle^{\perp})$ ,

\displaystyle\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{\mathsf{P}}_{ab}

\displaystyle=\frac{1}{n}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{\mathsf{Ric}}_{ab}=0\,,

i.e. $\widetilde{k}^{a}\widetilde{\mathsf{P}}_{ab}=\widetilde{f}\widetilde{g}_{ab}\widetilde{k}^{b}$ for some $\widetilde{k}$ , and so

	$\displaystyle\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}\widetilde{\mathsf{Y}}_{abc}$	$\displaystyle=\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}\left(\widetilde{\nabla}_{b}\widetilde{\mathsf{P}}_{ca}-\widetilde{\nabla}_{c}\widetilde{\mathsf{P}}_{ba}\right)$
		$\displaystyle=-2(\widetilde{v}^{b}\widetilde{\nabla}_{b}\widetilde{k}^{a})\widetilde{k}^{c}\widetilde{\mathsf{P}}_{ca}+(\widetilde{k}^{c}\widetilde{\nabla}_{c}\widetilde{k}^{a})\widetilde{v}^{b}\widetilde{\mathsf{P}}_{ba}+(\widetilde{k}^{c}\widetilde{\nabla}_{c}\widetilde{v}^{b})\widetilde{k}^{a}\widetilde{\mathsf{P}}_{ba}$
		$\displaystyle=-2(\widetilde{v}^{b}\widetilde{\nabla}_{b}\widetilde{k}^{a})\widetilde{f}\widetilde{g}_{ac}\widetilde{k}^{c}+(\widetilde{k}^{c}\widetilde{\nabla}_{c}\widetilde{k}^{a})\widetilde{v}^{b}\widetilde{\mathsf{P}}_{ba}+(\widetilde{k}^{c}\widetilde{\nabla}_{c}\widetilde{v}^{b})\widetilde{f}\widetilde{g}_{ab}\widetilde{k}^{b}\,.$

The first term is clearly zero, while the last two terms must vanish since $\widetilde{k}^{a}$ is tangent to geodesic curves. This establishes the result. ∎

We can weaken our assumption by replacing the twist-induced nearly Robinson geometry by an optical geometry $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ of dimension $2m+2>4$ with twisting non-shearing congruence of null geodesics $\widetilde{\mathcal{K}}$ , together with the Weyl curvature condition (5.8). By Theorem 3.1, this implies that the twist of $\widetilde{\mathcal{K}}$ induces a nearly Robinson structure $(\widetilde{N},\widetilde{k})$ with distinguished generator $\widetilde{k}$ of $\widetilde{K}$ , and $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{N},\widetilde{k})$ is locally fibred over an almost CR manifold $(\mathcal{M},H,J)$ . In addition, each $\widetilde{g}_{\theta}\in\accentset{n.e.}{\widetilde{\mathbf{c}}}$ for some pseudo-hermitian structure $\theta$ with Levi form $h$ takes the form (3.8) for some adapted coframe. In fact, we shall assume that the stronger condition on Weyl tensor of $\widetilde{\mathbf{c}}$

\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}

\displaystyle=0\,,

for any

\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})

(6.11)

Now, let $\widetilde{\sigma}$ be an almost Lorentzian scale, and suppose that $\widetilde{\sigma}$ satisfies (6.10). Then by Proposition 6.1, $\widetilde{\sigma}$ has non-empty zero set $\widetilde{\mathcal{Z}}$ , and off $\widetilde{\mathcal{Z}}$ , the metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ takes the form

\displaystyle\widetilde{g}

\displaystyle=\sec^{2}\phi\cdot\widetilde{g}_{\theta}\,,

for some metric $\widetilde{g}_{\theta}\in\accentset{n.e.}{\widetilde{\mathbf{c}}}$ , which can be taken to be of the form (3.8). By virtue of (6.10), the Ricci tensor of $\widetilde{g}$ satisfies $\widetilde{\mathsf{Ric}}(\widetilde{k},\widetilde{v})=0$ for all $\widetilde{v}\in\Gamma(\langle\widetilde{k}\rangle^{\perp})$ . It is shown in [TC22] that this Ricci curvature condition together with (6.11) is equivalent to $\widetilde{\lambda}_{\alpha}=0$ . In conclusion:

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ be an optical geometry of dimension $2m+2>4$ with twisting non-shearing congruence of null geodesics $\widetilde{\mathcal{K}}$ . Suppose that the Weyl tensor satisfies (6.11) and $\widetilde{\bm{c}}$ admits an almost Lorentzian scale $\widetilde{\sigma}$ that satisfies (6.10). Then the leaf space of $\widetilde{\mathcal{K}}$ is an almost CR manifold induced by the twist of $\widetilde{\mathcal{K}}$ , and the metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ takes the form

\displaystyle\widetilde{g}

\displaystyle=\sec^{2}\phi\cdot\left(4\theta\odot\left(\mathrm{d}\phi+\widetilde{\lambda}_{0}\theta\right)+h\right)\,,

(6.12)

for some smooth function $\widetilde{\lambda}_{0}$ , and where $\theta$ is a pseudo-hermitian structure with Levi form $h$ , and $\phi$ an affine parameter along $\widetilde{\mathcal{K}}$ .

We can check that this is consistent with Proposition 5.8, Remark 5.9 and Lemma 6.6.

Suppose now that $\widetilde{\sigma}$ is an almost weakly half-Einstein scale, i.e. it satisfies (3.13). By Proposition 3.10, off $\widetilde{\mathcal{Z}}$ , the Ricci tensor satisfies $\widetilde{\mathsf{Ric}}(\widetilde{v},\widetilde{v})=0$ for all $\widetilde{v}\in\Gamma(\widetilde{N})$ , which is shown to be equivalent in [TC22] to $\widetilde{\lambda}_{\alpha}=0$ , and the pseudo-hermitian invariants satisfy $\mathsf{A}_{\alpha\beta}=\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}=0$ . An argument similar to the proof of Theorem 2.4 allows us to conclude:

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ be an optical geometry of dimension $2m+2>4$ with twisting non-shearing congruence of null geodesics $\widetilde{\mathcal{K}}$ . Suppose that the Weyl tensor satisfies (6.11) and $\widetilde{\bm{c}}$ admits an almost weakly half-Einstein scale. Then the leaf space of $\widetilde{\mathcal{K}}$ is an almost CR manifold induced by the twist of $\widetilde{\mathcal{K}}$ , the metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ takes the form (6.12), and the CR scale $s$ corresponding to $\theta$ satisfies


	$\displaystyle\nabla_{(\alpha}\nabla_{\beta)}s+\mathrm{i}\mathsf{A}_{\alpha\beta}s+\mathsf{N}_{\gamma(\alpha\beta)}\nabla^{\gamma}s=0\,,$		(6.13a)
	$\displaystyle\nabla^{\gamma}s\mathsf{N}_{\gamma(\alpha\beta)}+\frac{1}{m}\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}s=0\,.$		(6.13b)

Finally, suppose that $\widetilde{\sigma}$ is an almost half-Einstein scale, i.e. it satisfies (3.13) and (3.14). Then, in particular, statement (6.9) of Proposition 6.9 must hold. By Proposition 3.10, condition (3.14) tells us that, off $\widetilde{\mathcal{Z}}$ , the Ricci scalar is constant, which is shown in [TC23] to be equivalent to a second-order differential equation on $\widetilde{\lambda}_{0}$ , for $\phi\neq\tfrac{2k+1}{2}\pi$ , $k\in\mathbf{Z}$ ,

\ddot{\widetilde{\lambda}}_{0}+2(2m+1)\tan\phi\dot{\widetilde{\lambda}}_{0}+(-4m(m+1)+2(m+1)(2m+1)\sec^{2}\phi)\widetilde{\lambda}_{0}\\ -2(m+1)\widetilde{\Lambda}\sec^{2}\phi+2m\Lambda=0\,,

(6.14)

where $\Lambda=\frac{1}{2}\widetilde{\mathsf{Sc}}$ is constant, $\widetilde{\mathsf{Sc}}$ being the Ricci scalar of $\widetilde{g}$ , and $\Lambda:=\frac{1}{m}\left(\mathsf{Sc}-\|\mathsf{N}\|^{2}\right)$ , with $\mathsf{Sc}$ being the Webster–Ricci scalar of $\theta$ . Solving for $\widetilde{\lambda}_{0}$ proves:

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ be an optical geometry of dimension $2m+2>4$ with twisting non-shearing congruence of null geodesics $\widetilde{\mathcal{K}}$ . Suppose that the Weyl tensor satisfies (6.11) and $\widetilde{\bm{c}}$ admits almost weakly half-Einstein scale. Then the leaf space of $\widetilde{\mathcal{K}}$ is an almost CR manifold induced by the twist of $\widetilde{\mathcal{K}}$ , the metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ takes the form (6.12), the CR scale $s$ corresponding to $\theta$ satisfies (6.13), and $\widetilde{\lambda}_{0}$ is given by

\displaystyle\widetilde{\lambda}_{0}

\displaystyle=\sum_{-m-1}^{m+1}\lambda_{0}^{(2k)}\mathrm{e}^{2\mathrm{i}k\phi}\,,

(6.15)

where


$\displaystyle\lambda_{0}^{(2m+2)}$	$\displaystyle=\frac{m!(m+1)!}{2(2m+2)!}\left((2m+1)\Lambda-(2m+2)\widetilde{\Lambda}\right)+\mu\,,$		(6.16a)
$\displaystyle\lambda_{0}^{(2k)}$	$\displaystyle=\frac{2(2m+1)!}{(m+1-k)!(m+1+k)!}\left(k\lambda_{0}^{(2m+2)}+(m+1-k)\Re(\mu)\right)\,,$	$\displaystyle 1\leq\|k\|\leq m+1\,,$	(6.16b)
$\displaystyle\lambda_{0}^{(0)}$	$\displaystyle=\frac{1}{2m+2}\Lambda+\frac{2(2m+1)!}{m!(m+1)!}\Re(\mu)\,,$		(6.16c)

where $\mu$ is some complex-valued smooth function on $\mathcal{M}$ , and $\Lambda:=\frac{1}{m}\left(\mathsf{Sc}-\|\mathsf{N}\|^{2}\right)$ , with $\mathsf{Sc}$ being the Webster–Ricci scalar of $\theta$ .

Since all the components $\widetilde{\lambda}_{\alpha}$ and $\widetilde{\lambda}_{0}$ are determined as Fourier exansion, applying Lemma 5.7 results in the following:

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ be an optical geometry of dimension $2m+2>4$ with twisting non-shearing congruence of null geodesics $\widetilde{\mathcal{K}}$ . Suppose that the Weyl tensor satisfies (6.11) and $\widetilde{\mathbf{c}}$ admits an almost half-Einstein scale $\widetilde{\sigma}$ . Then $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ is locally conformally isometric to a perturbed Fefferman space.

The results of the previous sections can easily be adapted to the case where the optical geometry under consideration is a perturbed Fefferman space. In this section, we describe perturbed Fefferman spaces admitting almost half-Einstein and Einstein scales.

\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}

\displaystyle=0\,,

for any

\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})

(6.17)

Then $\widetilde{\mathbf{c}}_{\widetilde{\xi}}$ admits an almost half-Einstein scale $\widetilde{\sigma}$ , if and only if $\widetilde{\xi}$ is determined by some CR data $\left(\bm{\xi}_{\alpha}^{(0)},[\nabla,\bm{\xi}_{0}^{(0)}],\bm{\xi}_{0}^{(2k)}\right)_{k=1,\ldots,m+1}$ , and there exists a non-vanishing density $\sigma\in\Gamma(\mathcal{E}(1,0))$ such that the CR scale $s:=\sigma\overline{\sigma}$ satisfies


	$\displaystyle\nabla_{(\alpha}\nabla_{\beta)}s+\mathrm{i}\mathsf{A}_{\alpha\beta}s+\mathsf{N}_{\gamma(\alpha\beta)}\nabla^{\gamma}s=0\,,$		(6.18a)
	$\displaystyle\nabla^{\gamma}s\mathsf{N}_{\gamma(\alpha\beta)}+\frac{1}{m}\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}s=0\,,$		(6.18b)

and


$\displaystyle\bm{\xi}_{\alpha}^{(0)}$	$\displaystyle=-\mathrm{i}\sigma^{-1}\nabla_{\alpha}\sigma\,,$
$\displaystyle\bm{\xi}_{0}^{(2m+2)}$	$\displaystyle=\left(\frac{m!(m+1)!}{2(2m+2)!}\left((2m+1)\Lambda-(2m+2)\widetilde{\Lambda}\right)+\mu\right)\sigma^{m}\overline{\sigma}^{-m-2}\,,$
$\displaystyle\bm{\xi}_{0}^{(2k)}$	$\displaystyle=\frac{2(2m+1)!}{(m+1-k)!(m+1+k)!}\left(k\bm{\xi}_{0}^{(2m+2)}+(m+1-k)\Re(\mu)\right)\sigma^{k-1}\overline{\sigma}^{-k-1}\,,$	$\displaystyle 1\leq\|k\|\leq m+1\,,$
$\displaystyle\bm{\xi}_{0}^{(0)}$	$\displaystyle=\frac{\mathrm{i}}{m}\left(\nabla_{\alpha}\bm{\xi}^{\alpha}_{(0)}-\nabla^{\alpha}\bm{\xi}_{\alpha}^{(0)}\right)+\frac{2(2m+1)!}{m!(m+1)!}\Re(\mu)\sigma^{-1}\overline{\sigma}^{-1}\,,$

for some real constant $\widetilde{\Lambda}$ , real function $\Lambda:=\frac{1}{m}\left(\mathsf{Sc}-\|\mathsf{N}\|^{2}\right)$ , and complex-valued function $\mu$ on $(\mathcal{M},H,J)$ .

This being the case, the zero set $\widetilde{\mathcal{Z}}$ of $\widetilde{\sigma}$ consists of the union $\widetilde{\mathcal{Z}}_{+}\cup\widetilde{\mathcal{Z}}_{-}$ of the sections $\widetilde{\mathcal{Z}}_{\pm}=\{[\pm\mathrm{i}\sigma^{-1}]:\mathcal{M}\rightarrow\widetilde{\mathcal{M}}\}$ . Off $\widetilde{\mathcal{Z}}$ , the half-Einstein metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ has Ricci scalar $(2m+2)\widetilde{\Lambda}$ , and is given $\widetilde{g}=\sec^{2}\phi\cdot\widetilde{g}_{\theta,\widetilde{\xi}}$ , where $\widetilde{g}_{\theta,\widetilde{\xi}}$ is the perturbed Fefferman metric associated to $\sigma$ and contact form $\theta$ , and $\phi$ the local fibre coordinate

This is really a corollary of Proposition 6.10 where we take our optical geometry to be a perturbed Fefferman space in the first place. We begin by interpreting the coordinate $\phi$ as being defined by a distinguished non-vanishing density $\sigma$ of weight $(1,0)$ as in Proposition 6.4, which is related to the CR scale as $s=\sigma\overline{\sigma}$ . We can determine the components of the perturbation one-form $\widetilde{\xi}$ by expressing the perturbed Fefferman metric as (5.5), and we finally turn these into densities by applying Lemma 5.2 — see relations (5.2). The converse works by reversing the steps. ∎

Our next step would be to consider almost pure radiation scales, however, the following lemma, which is a straightforward reformulation of a result in [TC22], will give us a short-cut to almost Einstein scales.

An almost pure radiation scale on a perturbed Fefferman space of dimension $2m+2>4$ whose Weyl tensor satisfies (6.17) is necessarily an almost Einstein scale.

We can presently finish this section with a description of perturbed Fefferman spaces admitting almost Einstein scales.

\displaystyle\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{a}\widetilde{v}^{b}\widetilde{k}^{c}

\displaystyle=0\,,

for any

\widetilde{k}\in\Gamma(\widetilde{K}),\widetilde{v}\in\Gamma(\widetilde{K}^{\perp})

and $\widetilde{\mathbf{c}}_{\widetilde{\xi}}$ admits an almost Einstein scale $\widetilde{\sigma}$ . Then $\widetilde{\xi}$ is determined by CR data $\left(\bm{\xi}_{\alpha}^{(0)},[\nabla,\bm{\xi}_{0}^{(0)}],\bm{\xi}_{0}^{(2k)}\right)_{k=1,\ldots,m+1}$ and there exists a non-vanishing density $\sigma\in\Gamma(\mathcal{E}(1,0))$ such that $s:=\sigma\overline{\sigma}$ is an almost CR–Einstein scale, i.e. $s$ satisfies


	$\displaystyle\nabla_{(\alpha}\nabla_{\beta)}s+\mathrm{i}\mathsf{A}_{\alpha\beta}s+\mathsf{N}_{\gamma(\alpha\beta)}\nabla^{\gamma}s=0\,,$		(6.20a)
	$\displaystyle\nabla^{\gamma}s\mathsf{N}_{\gamma(\alpha\beta)}+\frac{1}{m}\nabla^{\gamma}\mathsf{N}_{\gamma(\alpha\beta)}s=0\,,$		(6.20b)
	$\displaystyle\left(s\nabla_{\bar{\beta}}\nabla_{\alpha}s-\nabla_{\alpha}s\nabla_{\bar{\beta}}s+\mathsf{P}_{\alpha\bar{\beta}}s^{2}-\frac{1}{m+2}\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}s^{2}\right)_{\circ}=0\,.$		(6.20c)

and


$\displaystyle\bm{\xi}_{\alpha}^{(0)}$	$\displaystyle=-\mathrm{i}\sigma^{-1}\nabla_{\alpha}\sigma\,,$		(6.21a)
$\displaystyle\bm{\xi}_{0}^{(2m+2)}$	$\displaystyle=\left(\frac{m!(m+1)!}{2(2m+2)!}\left((2m+1)\Lambda-(2m+2)\widetilde{\Lambda}\right)+\mu\right)\sigma^{m}\overline{\sigma}^{-m-2}\,,$		(6.21b)
$\displaystyle\bm{\xi}_{0}^{(2k)}$	$\displaystyle=\frac{2k(2m+1)!}{(m+1-k)!(m+1+k)!}\bm{\xi}_{0}^{(2m+2)}\sigma^{k-1}\overline{\sigma}^{-k-1}\,,$	$\displaystyle 1\leq\|k\|\leq m+1\,,$	(6.21c)
$\displaystyle\bm{\xi}_{0}^{(0)}$	$\displaystyle=\frac{\mathrm{i}}{m}\left(\nabla_{\alpha}\bm{\xi}^{\alpha}_{(0)}-\nabla^{\alpha}\bm{\xi}_{\alpha}^{(0)}\right)\,,$		(6.21d)

for some real constants $\widetilde{\Lambda}$ and $\Lambda:=\frac{1}{m}\left(\mathsf{Sc}-\|\mathsf{N}\|^{2}\right)$ , and complex-valued function $\mu$ on $(\mathcal{M},H,J)$ satisfying $\overline{\mu}=-\mu$ .

This being the case, the zero set $\widetilde{\mathcal{Z}}$ of $\widetilde{\sigma}$ consists of the union $\widetilde{\mathcal{Z}}_{+}\cup\widetilde{\mathcal{Z}}_{-}$ of the sections $\widetilde{\mathcal{Z}}_{\pm}=\{[\pm\mathrm{i}\sigma^{-1}]:\mathcal{M}\rightarrow\widetilde{\mathcal{M}}\}$ . Off $\widetilde{\mathcal{Z}}$ , the Einstein metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ has Ricci scalar $(2m+2)\widetilde{\Lambda}$ , and is given $\widetilde{g}=\sec^{2}\phi\cdot\widetilde{g}_{\theta,\widetilde{\xi}}$ , where $\widetilde{g}_{\theta,\widetilde{\xi}}$ is the perturbed Fefferman metric associated to $\sigma$ and contact form $\theta$ , and $\phi$ the local fibre coordinate.

Our assumption that $\widetilde{\sigma}$ is an almost Einstein scale clearly allows us to re-use the setting of Proposition 6.10 and Theorem 6.12 to which this proof will refer. The only difference is that there are additional constraints on the Ricci tensor of the metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ . First, the condition $\widetilde{\mathsf{Ric}}(\widetilde{v},\widetilde{w})=\widetilde{\Lambda}\widetilde{g}(\widetilde{v},\widetilde{w})$ for any $\widetilde{v}\in\Gamma(\langle\widetilde{k}\rangle^{\perp})$ , $\widetilde{w}\in\Gamma(\widetilde{N})$ has the following two consequences:

•

The pseudo-hermitian structure $\theta$ is CR–Einstein, and so, by Theorem 2.4, the corresponding CR scale must satisfy (6.20) in agreement with Theorem 6.12. This also means that $\Lambda:=\frac{1}{m}\left(\mathsf{Sc}-\|\mathsf{N}\|^{2}\right)$ is now constant.

•

The complex-valued function $\widetilde{\lambda}_{0}$ is subject to the ordinary differential equation

\displaystyle\tan\phi\dot{\widetilde{\lambda}}_{0}-(2(m+1)-(2m+1)\sec^{2}\phi)\widetilde{\lambda}_{0}-\widetilde{\Lambda}\sec^{2}\phi+\Lambda=0\,,

(6.22)

in addition to (6.14). Plugging the solution (6.15) of (6.14) with (6.16) into (6.22) determines $\mu$ as a purely imaginary function, i.e. $\Re(\mu)=0$ , which in comparison with the form of the CR data given in Theorem 6.12, yields (6.21).

∎

This result tells us that in general, a Fefferman conformal structure (i.e. with $\widetilde{\xi}=0$ ) cannot admit an almost Einstein scale unless the underlying CR–Einstein structure arises from a complex density $\sigma$ satisfying equations (2.17), and $(2m+1)\Lambda-(2m+2)\widetilde{\Lambda}=\mu=0$ .

Let $\widetilde{\sigma}$ be an almost Lorentzian scale on a perturbed Fefferman space $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}}_{\widetilde{\xi}},\widetilde{k})\longrightarrow(\mathcal{M},H,J)$ of dimension $2m+2\geq 4$ . Then, for each choice of Levi-Civita connection $\widetilde{\nabla}$ , we have a weighted one-form

\displaystyle\widetilde{\bm{\nu}}_{a}

\displaystyle:=\widetilde{\nabla}_{a}\widetilde{\sigma}\in\Gamma(\widetilde{\mathcal{E}}_{a}[1])\,.

(7.1)

This depends on the choice of metric in $\widetilde{\mathbf{c}}_{\widetilde{\xi}}$ since under a conformal change, $\widetilde{\bm{\nu}}_{a}$ transforms as $\widehat{\widetilde{\bm{\nu}}}_{a}=\widetilde{\bm{\nu}}_{a}+\widetilde{\Upsilon}_{a}\widetilde{\sigma}$ . However, on restriction to $\widetilde{\mathcal{Z}}$ , any of these weighted forms coincides with the weighted normal covector of $\widetilde{\mathcal{Z}}$ [CG18].

Let us assume further that $\widetilde{\sigma}$ is an almost half-Einstein scale. By Theorem 6.12, $\widetilde{\sigma}$ is determined by some non-vanishing $\sigma\in\Gamma(\mathcal{E}(1,0))$ trivialising $\widetilde{\mathcal{M}}\rightarrow\mathcal{M}$ with fibre coordinate $\phi$ and with contact form $\theta=(\sigma\overline{\sigma})^{-1}\bm{\theta}$ , associated Fefferman scale $\widetilde{\sigma}_{\theta,\widetilde{\xi}}$ so that $\widetilde{\sigma}=\cos\phi\cdot\widetilde{\sigma}_{\theta,\widetilde{\xi}}$ . Its zero set is given by $\widetilde{\mathcal{Z}}=\widetilde{\mathcal{Z}}_{+}\cup\widetilde{\mathcal{Z}}_{-}$ where $\widetilde{\mathcal{Z}}_{\pm}=\{[\pm\mathrm{i}\sigma^{-1}]:\mathcal{M}\rightarrow\widetilde{\mathcal{M}}\}$ . This means two things: First, for each choice of Levi-Civita connection $\widetilde{\nabla}$ , the weighted one-form (7.1) reads as

\displaystyle\widetilde{\bm{\nu}}

\displaystyle=-\widetilde{\sigma}_{\theta,\widetilde{\xi}}\cdot\sin\phi\cdot\mathrm{d}\phi+\cos\phi\cdot\widetilde{\nabla}\widetilde{\sigma}_{\theta,\widetilde{\xi}}\,.

Second, we may choose our Levi-Civita connection to preserve the distinguished perturbed Fefferman scale $\widetilde{\sigma}_{\theta,\widetilde{\xi}}$ to single out the one-form of weight $1$ and vector field of weight $-1$

\displaystyle\accentset{(\theta)}{\widetilde{\bm{\nu}}}

\displaystyle=-\widetilde{\sigma}_{\theta,\widetilde{\xi}}\cdot\sin\phi\cdot\mathrm{d}\phi\,,

\displaystyle\accentset{(\theta)}{\widetilde{\bm{n}}}

\displaystyle=\widetilde{\bm{g}}^{-1}(\accentset{(\theta)}{\widetilde{\bm{\nu}}},\cdot)\,.

Computing the length squared of $\accentset{(\theta)}{\widetilde{\bm{n}}}$ on restriction to $\widetilde{\mathcal{Z}}$ , we find

\displaystyle\widetilde{\bm{g}}(\accentset{(\theta)}{\widetilde{\bm{n}}},\accentset{(\theta)}{\widetilde{\bm{n}}})|_{\widetilde{\mathcal{Z}}}

\displaystyle=-\frac{1}{2m+1}\widetilde{\Lambda}\,.

We see that $\widetilde{\mathcal{Z}}$ is spacelike, timelike or null if its normal is timelike, spacelike or null respectively. Hence, the previous display immediately yields the following proposition (see also \citesPenrose1965,Penrose1986 in the four-dimensional case).

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}}_{\widetilde{\xi}},\widetilde{k})\longrightarrow(\mathcal{M},H,J)$ be a perturbed Fefferman space of dimension $2m+2$ that admits an almost half-Einstein scale $\widetilde{\sigma}$ , so that off the zero set $\widetilde{\mathcal{Z}}$ of $\widetilde{\sigma}$ , the metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ is half-Einstein with constant Ricci scalar $(2m+2)\widetilde{\Lambda}$ . Then

•

$\widetilde{\mathcal{Z}}$ is null if and only if $\widetilde{\Lambda}=0$ ;
•

$\widetilde{\mathcal{Z}}$ is spacelike if and only if $\widetilde{\Lambda}>0$ ;
•

$\widetilde{\mathcal{Z}}$ is timelike if and only if $\widetilde{\Lambda}<0$ .

In general, while $\widetilde{\mathcal{Z}}$ is expected to have merely a conformal, rather than metric, structure — see for instance [CG18] — in our case, by virtue of the almost half-Einstein equations, we do in fact have a distinguished regular metric on $\widetilde{\mathcal{Z}}$ , namely, the perturbed Fefferman metric pulled back to $\widetilde{\mathcal{Z}}$ :

\displaystyle\widetilde{g}_{\theta,\widetilde{\xi}}|_{\widetilde{\mathcal{Z}}}

\displaystyle=4\theta\odot\left(\frac{\mathrm{i}}{2}\left(\sigma^{-1}\nabla\sigma-\overline{\sigma}^{-1}\nabla\overline{\sigma}\right)-\left(\frac{1}{m+2}\mathsf{P}+\frac{1}{2m(m+1)}\|\mathsf{N}\|^{2}\right)\theta+\widetilde{\xi}|_{\widetilde{\mathcal{Z}}}\right)+h\,.

This clearly depends on the pseudo-hermitian data. Proposition 7.1 can also derived by computing the determinant $\det\left(\left.\widetilde{g}_{\theta,\widetilde{\xi}}\right|_{\widetilde{\mathcal{Z}}}\right)=-\frac{4}{2m+1}\widetilde{\Lambda}$ . When $\accentset{(\theta)}{\widetilde{\bm{n}}}$ is null, it is tangent to $\widetilde{\mathcal{Z}}$ and the metric $\widetilde{g}_{\theta,\widetilde{\xi}}$ is degenerate.

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}}_{\widetilde{\xi}},\widetilde{k})\rightarrow(\mathcal{M},H,J)$ be a perturbed Fefferman space of dimension $2m+2>4$ that admits an almost Einstein scale $\widetilde{\sigma}$ with zero set $\widetilde{\mathcal{Z}}$ , so that off the zero set $\widetilde{\mathcal{Z}}$ of $\widetilde{\sigma}$ , the metric $\widetilde{g}=\widetilde{\sigma}^{-2}\widetilde{\bm{g}}$ is Einstein with constant Ricci scalar $(2m+2)\widetilde{\Lambda}$ . Then $\widetilde{\mathbf{c}}_{\widetilde{\xi}}$ is conformally flat on $\widetilde{\mathcal{Z}}$ if and only if $(\mathcal{M},H,J)$ is CR flat and the CR–Einstein structure has Webster–Ricci scalar $\frac{m(2m+2)}{2m+1}\widetilde{\Lambda}$ .

This follows from a direct computation. The Weyl curvature of the perturbed Fefferman metric is given by the set of equations (8.24) of reference [TC22]. Evaluating these at $\phi=\pm\frac{\pi}{2}$ gives us the Weyl curvature on $\widetilde{\mathcal{Z}}$ , it is then straightforward, if not somewhat tedious, to check that the Weyl tensor vanishes on $\widetilde{\mathcal{Z}}$ if and only if $\mathsf{N}_{\alpha\beta\gamma}=0$ , $\mathsf{S}_{\alpha\bar{\beta}\gamma\bar{\delta}}=0$ , i.e. $(\mathcal{M},H,J)$ is CR flat, and $\frac{\Lambda}{2m+2}-\frac{\widetilde{\Lambda}}{2m+1}=0$ . ∎

The above result should be contrasted with its four-dimensional counterpart given in [TC23].

Given a perturbed Fefferman space $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}}_{\widetilde{\xi}},\widetilde{k})\longrightarrow(\mathcal{M},H,J)$ , one may naturally ask how conformal symmetries of $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}}_{\widetilde{\xi}},\widetilde{k})$ relate to solutions to CR invariant differential equations on its base $(\mathcal{M},H,J)$ . While this question goes beyond the scope of this article, we shall nevertheless outline some ideas towards the answer.

Let us first review the situation in the unperturbed case, i.e. $\widetilde{\xi}=0$ and when $(\mathcal{M},H,J)$ is a CR manifold. It is shown in [ČG08] that the space of all conformal Killing fields on a Fefferman space $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{k})$ splits into a direct sum of three spaces

\displaystyle\widetilde{\mathcal{A}}\oplus\widetilde{\mathcal{B}}\oplus\langle\widetilde{k}\rangle\,,

(8.1)

where

(1)

any section of $\widetilde{\mathcal{A}}$ is transverse to $\langle\widetilde{k}\rangle^{\perp}$ and arises from a real solution $s\in\Gamma(\mathcal{E}(1,1))$ of

$\displaystyle\nabla_{\alpha}\nabla_{\beta}s+\mathrm{i}\mathsf{A}_{\alpha\beta}s$ $\displaystyle=0\,,$

and such a solution gives rise to transverse infinitesimal CR symmetries as we have already seen in Section 2;

(2)

any section of $\widetilde{\mathcal{B}}$ is tangent to $\langle\widetilde{k}\rangle^{\perp}$ , but not tangent to $\langle\widetilde{k}\rangle$ , and arises from a solution $w^{\alpha}\in\Gamma(\mathcal{E}^{\alpha}(-1,1))$ to the CR invariant differential equations

\displaystyle\nabla^{(\alpha}w^{\beta)}

\displaystyle=0\,,

\displaystyle\nabla_{\alpha}w^{\beta}

\displaystyle=\frac{1}{m}\delta_{\alpha}^{\beta}\nabla_{\gamma}w^{\gamma}\,.

In dimension three, however, the second condition is vacuous, while the first one reduces to $\nabla^{\alpha}w^{\beta}=0$ .

For conformally flat $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{k})$ , the dimensions of $\widetilde{\mathcal{A}}$ and $\widetilde{\mathcal{B}}$ are $(m+1)(m+3)$ and $(m+1)(m+2)$ respectively, the former being the maximal dimension of the automorphism group of $(\mathcal{M},H,J)$ as expected. Together with $\langle\widetilde{k}\rangle$ , these indeed add up to $(m+2)(2m+3)$ , the maximal dimension of the automorphism group of $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{k})$ .

There are two ways in which a perturbation of $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{k})$ by a semi-basic one-form $\widetilde{\xi}$ affects the decomposition (8.1): first, $\widetilde{k}$ itself will cease to be conformal Killing (unless $\widetilde{k}\makebox[7.0pt]{\rule{6.0pt}{0.3pt}\rule{0.3pt}{5.0pt}}\,\mathrm{d}\widetilde{\xi}=0$ ). Second, the perturbation one-form will affect the conformal curvature, and in particular, the dimension of each summand in (8.1) will not be preserved.

Interestingly, if we start with a conformally flat $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{k})$ , then we end up with non-conformally flat perturbed Fefferman spaces over a flat CR manifold. To illustrate this phenomenon in dimension four, and following [Tra02] a flat CR three-manifold (i.e. either the Heisenberg group or three-sphere) gives rise to four distinct perturbed Fefferman spaces, each admitting an almost Einstein scale $\widetilde{\sigma}$ with zero set $\widetilde{\mathcal{Z}}$ :

•

the canonical conformally flat Fefferman conformal structure that contains the Minkowski metric off $\widetilde{\mathcal{Z}}$ ;
•

a perturbed Fefferman conformal structure that contains the Petrov type D Taub-NUT metric \citesTaub1951,Newman1963 off $\widetilde{\mathcal{Z}}$ ;
•

a perturbed Fefferman conformal structure that contains the Petrov type N Hauser metric [Hau74] off $\widetilde{\mathcal{Z}}$ .

Incidentally, since in dimension four, the ‘unperturbed’ Fefferman conformal structure admits an almost Einstein scale if and only if it is conformally flat (or equivalently its underlying CR structure is flat), the main point of these perturbations is to enlarge the range of possible almost Einstein scales.

In fact, a four-dimensional conformally flat space $(\mathcal{M},\widetilde{\mathbf{c}})$ can be viewed as a (perturbed) Fefferman space in ‘infinitely many’ ways: The space of all null geodesics in $\widetilde{\mathcal{M}}$ is a contact CR five-fold $\mathbb{PN}$ of signature $(1,1)$ , and any contact CR submanifold of $\mathbb{PN}$ of dimension three gives rise to a twisting non-shearing congruences of null geodesics on $\widetilde{\mathcal{M}}$ . In the analytic category, the so-called Kerr theorem tells us that such CR three-manifolds can be constructed as the intersection of $\mathbb{PN}\subset\mathbf{C}\mathbb{P}^{3}$ with a complex surface in $\mathbf{C}\mathbb{P}^{3}$ , the twistor space of $(\mathcal{M},\widetilde{\mathbf{c}})$ [Pen67, PR86]. For instance, the ‘massless’ Kerr metric is flat, but its underlying CR three-manifold admits a two-dimensional group of symmetry.

In [LN90], the authors consider algebraically special pure radiation metrics, which can therefore be treated as almost pure radiation scales on a perturbed Fefferman space, with underlying contact CR three-manifold admitting an automorphism group of submaximal dimension (in this case, three).

When $(\mathcal{M},H,J)$ is a strictly almost CR manifold, one would expect the decomposition (8.1) to carry over, but would certainly have to change the interpretation of the bundle $\widetilde{\mathcal{A}}$ , and possibly $\widetilde{\mathcal{B}}$ . Then there is the added complication of the parameter $\alpha$ in Definitions 4.1 and 4.2. We shall not attempt to answer these questions at this stage.

It is shown in [TC22] that locally the anti-canonical bundle of any almost Kähler–Einstein manifold admits an almost CR–Einstein manifold, and any almost CR–Einstein manifold locally arises in this way. This allows us to produce examples of almost CR–Einstein manifolds. For the integrable case, see Leitner [Lei07]. Based on this remark, we refer to the explicit construction of a strictly almost Kähler–Einstein four-manifold by Nurowski and Przanowski [NP99] to construct an example of an strictly almost CR–Einstein manifold.

Consider the subset of $\mathbf{R}^{5}$ with coordinates $(u,z^{\alpha},\overline{z}^{\bar{\alpha}})_{\alpha=1,2}$ with $z^{1}+\overline{z}^{\bar{1}}-2z^{2}\overline{z}^{\bar{2}}>0$ , and define

$\displaystyle\theta$	$\displaystyle=\mathrm{d}u+\mathrm{i}z^{1}\mathrm{d}z^{2}-\mathrm{i}z^{2}\mathrm{d}z^{1}-\mathrm{i}\overline{z}^{\bar{1}}\mathrm{d}\overline{z}^{\bar{2}}+\mathrm{i}\overline{z}^{\bar{2}}\mathrm{d}\overline{z}^{\bar{1}}\,,$
$\displaystyle\theta^{1}$	$\displaystyle=f^{-\frac{1}{4}}\cdot\left(\mathrm{d}z^{1}-2\overline{z}^{\bar{2}}\cdot\mathrm{d}z^{2}+f^{\frac{1}{2}}\cdot\mathrm{d}\overline{z}^{\bar{2}}\right)\,,$	$\displaystyle\overline{\theta}^{\bar{1}}$	$\displaystyle=\overline{\theta^{1}}\,,$
$\displaystyle\theta^{2}$	$\displaystyle=f^{-\frac{1}{4}}\cdot\left(-\mathrm{d}\overline{z}^{\bar{1}}+2z^{2}\cdot\mathrm{d}\overline{z}^{\bar{2}}+f^{\frac{1}{2}}\cdot\mathrm{d}z^{2}\right)\,,$	$\displaystyle\overline{\theta}^{\bar{2}}$	$\displaystyle=\overline{\theta^{2}}\,,$

where

\displaystyle f

\displaystyle:=4\cdot(z^{1}+\overline{z}^{\bar{1}}-2z^{2}\overline{z}^{\bar{2}})\,.

Note that

\displaystyle\mathrm{d}f

\displaystyle=2f^{\frac{1}{4}}\cdot\left(\theta^{1}-\theta^{2}+\overline{\theta}^{\bar{1}}-\overline{\theta}^{\bar{2}}\right)\,.

Then

	$\displaystyle\mathrm{d}\theta$	$\displaystyle=\mathrm{i}\theta^{1}\wedge\overline{\theta}^{\bar{1}}+\mathrm{i}\theta^{2}\wedge\overline{\theta}^{\bar{2}}\,,$
	$\displaystyle\mathrm{d}\theta^{1}$	$\displaystyle=f^{-\frac{3}{4}}\left(-\frac{1}{2}\theta^{1}\wedge\theta^{2}-\frac{1}{2}\theta^{1}\wedge\overline{\theta}^{\bar{1}}+\frac{1}{2}\theta^{1}\wedge\overline{\theta}^{\bar{2}}+\overline{\theta}^{\bar{1}}\wedge\overline{\theta}^{\bar{2}}\right)\,,$
	$\displaystyle\mathrm{d}\theta^{2}$	$\displaystyle=f^{-\frac{3}{4}}\left(-\frac{1}{2}\theta^{1}\wedge\theta^{2}-\frac{1}{2}\theta^{2}\wedge\overline{\theta}^{\bar{1}}+\frac{1}{2}\theta^{2}\wedge\overline{\theta}^{\bar{2}}+\overline{\theta}^{\bar{1}}\wedge\overline{\theta}^{\bar{2}}\right)\,.$

so that the connection $1$ -form is given by

\displaystyle\Gamma_{1}{}^{1}

\displaystyle=\frac{1}{2}f^{-\frac{3}{4}}\left(\theta^{1}-\theta^{2}-\overline{\theta}^{\bar{1}}+\overline{\theta}^{\bar{2}}\right)\,,

\displaystyle\Gamma_{2}{}^{2}

\displaystyle=\frac{1}{2}f^{-\frac{3}{4}}\left(\theta^{1}-\theta^{2}-\overline{\theta}^{\bar{1}}+\overline{\theta}^{\bar{2}}\right)\,,

\displaystyle\Gamma_{1}{}^{2}

\displaystyle=\Gamma_{2}{}^{1}=0\,,

and the Webster torsion and Nijenhuis tensor by

	$\displaystyle\mathsf{A}_{\alpha\beta}$	$\displaystyle=0\,,$		(A.1)
	$\displaystyle\mathsf{N}_{121}=\mathsf{N}_{122}$	$\displaystyle=-f^{-\frac{3}{4}}\,,$

respectively. This implies that


$\displaystyle\mathsf{N}_{\gamma\delta 1}\mathsf{N}^{\gamma\delta}{}_{\bar{1}}$	$\displaystyle=\mathsf{N}_{\gamma\delta 1}\mathsf{N}^{\gamma\delta}{}_{\bar{2}}=\mathsf{N}_{\gamma\delta 2}\mathsf{N}^{\gamma\delta}{}_{\bar{1}}=\mathsf{N}_{\gamma\delta 2}\mathsf{N}^{\gamma\delta}{}_{\bar{2}}=2f^{-\frac{3}{2}}\,,$	(A.2a)
$\displaystyle\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}$	$\displaystyle=2f^{-\frac{3}{2}}h_{\alpha\bar{\beta}}\,,$	(A.2b)
$\displaystyle\mathsf{N}_{\alpha\beta\gamma}\mathsf{N}^{\alpha\beta\gamma}$	$\displaystyle=4f^{-\frac{3}{2}}\,.$	(A.2c)

A direct computation shows

\displaystyle\nabla_{\bar{\delta}}\mathsf{N}_{\gamma\alpha\beta}

\displaystyle=0\,.

(A.3)

To compute the curvature two-form $\Omega_{\alpha}{}^{\beta}$ of $\Gamma_{\alpha}{}^{\beta}$ , it is convenient to use the identity

\displaystyle\mathrm{d}(\theta^{1}-\theta^{2})

\displaystyle=-\frac{1}{2}f^{-\frac{3}{4}}(\theta^{1}-\theta^{2})\wedge(\overline{\theta}^{\bar{1}}-\overline{\theta}^{\bar{2}})\,,

from which we easily obtain

\displaystyle\Omega_{2}{}^{2}=\Omega_{1}{}^{1}

\displaystyle=f^{-\frac{3}{2}}(\theta^{1}-\theta^{2})\wedge(\overline{\theta}^{\bar{1}}-\overline{\theta}^{\bar{2}})\,,

\displaystyle\Omega_{2}{}^{1}=\Omega_{2}{}^{1}

\displaystyle=0\,,

i.e.

\displaystyle\mathsf{R}_{1\bar{1}\alpha}{}^{\beta}=\mathsf{R}_{2\bar{2}\alpha}{}^{\beta}

\displaystyle=f^{-\frac{3}{2}}\delta_{\alpha}^{\beta}\,,

\displaystyle\mathsf{R}_{1\bar{2}\alpha}{}^{\beta}=\mathsf{R}_{2\bar{1}\alpha}{}^{\beta}

\displaystyle=-f^{-\frac{3}{2}}\delta_{\alpha}^{\beta}\,,

and all other components vanish. In particular, the Webster–Ricci tensor is given by

\displaystyle\mathsf{Ric}_{\alpha\bar{\beta}}

\displaystyle=2f^{-\frac{3}{2}}h_{\alpha\bar{\beta}}\,,

\displaystyle\mathsf{Sc}

\displaystyle=4f^{-\frac{3}{2}}\,,

so that using (A.2b), we find

\displaystyle\mathsf{Ric}_{\alpha\bar{\beta}}-\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}

\displaystyle=0\,.

(A.4)

By (A.1), (A.3) and (A.4), we can now conclude that $(\mathcal{M},H,J)$ as defined above is an almost CR–Einstein manifold — see equations (2.12).

Since by (A.2),

\displaystyle 2\mathsf{N}_{1\gamma\delta}\mathsf{N}_{\bar{2}}{}^{\gamma\delta}-\mathsf{N}_{\gamma\delta 1}\mathsf{N}^{\gamma\delta}{}_{\bar{2}}=-2f^{-\frac{3}{2}}\,,

condition (2.17d) is not satisfied, which means that the almost CR–Einstein scale $s$ cannot be expressed as $s=\sigma\overline{\sigma}$ where $\sigma$ is a density of weight $(1,0)$ that satisfies $\nabla_{\bar{\alpha}}\sigma=0$ . We shall nevertheless show that there exists a density $\widehat{\sigma}$ of weight $(1,0)$ that satisfies $\nabla_{\bar{\alpha}}\widehat{\sigma}=0$ but not $s=\widehat{\sigma}\overline{\widehat{\sigma}}$ . To this end, we define

\displaystyle\sigma

\displaystyle:=\left(\theta\wedge\theta^{1}\wedge\theta^{2}\right)^{-\frac{1}{4}}\,,

which clearly satisfies $s=\sigma\widehat{\sigma}$ , and compute

\displaystyle\nabla\sigma

\displaystyle=\mathrm{i}\xi\otimes\sigma\,,

where

\displaystyle\xi

\displaystyle=-\frac{\mathrm{i}}{4}f^{-\frac{3}{4}}\left(\theta^{1}-\theta^{2}-\overline{\theta}^{\bar{1}}+\overline{\theta}^{\bar{2}}\right)\,.

Note in particular that

\displaystyle\mathrm{d}\xi

\displaystyle=-\mathrm{i}f^{-\frac{3}{2}}\left(\theta^{1}\wedge\overline{\theta}^{\bar{1}}-\theta^{1}\wedge\overline{\theta}^{\bar{2}}+\theta^{2}\wedge\overline{\theta}^{\bar{1}}+\theta^{2}\wedge\overline{\theta}^{\bar{2}}\right)\,,

the one-form $\xi$ cannot possibly be exact, which means that there is no density $\sigma^{\prime}$ of CR weight $(1,0)$ such that $\sigma^{\prime}\overline{\sigma^{\prime}}$ is the almost CR–Einstein scale and $\nabla_{\bar{\alpha}}\sigma^{\prime}=0$ as claimed earlier.

However, for any $k\in\mathbf{R}$ , we have

\displaystyle\nabla(f^{k}\sigma)

\displaystyle=f^{k-\frac{3}{4}}\left((2k+\tfrac{1}{4})(\theta^{1}-\theta^{2})+(2k-\tfrac{1}{4})(\overline{\theta}^{\bar{1}}-\overline{\theta}^{\bar{2}})\right)\otimes\sigma\,.

Hence, taking $k=\frac{1}{8}$ , we find that $\widehat{\sigma}:=f^{\frac{1}{8}}\sigma$ satisfies

\displaystyle\nabla\widehat{\sigma}

\displaystyle=\frac{1}{2}f^{-\frac{3}{4}}(\theta^{1}-\theta^{2})\otimes\widehat{\sigma}\,,

i.e.

\displaystyle\nabla_{\bar{\alpha}}\widehat{\sigma}

\displaystyle=0\,,

\displaystyle s

\displaystyle=f^{-\frac{1}{4}}\widehat{\sigma}\overline{\widehat{\sigma}}\,,

as required. It is also interesting to note that $\widehat{\sigma}\overline{\widehat{\sigma}}$ is also distinct from $\|\mathsf{N}\|^{-2}$ .

This example thus shows that almost CR manifolds may admit CR densities, i.e. densities of weight $(1,0)$ annihilated by the distribution $H^{(0,1)}$ .

We essentially follow the proof given in [Gra87] for the involutive case. We first recall that a null conformal Killing field $\widetilde{k}$ on $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}})$ can equivalently be expressed in terms of the weighted one-form $\widetilde{\bm{\kappa}}_{a}=\widetilde{\bm{g}}_{ab}\widetilde{k}^{b}$ solving the conformally invariant equation

\displaystyle\nabla_{a}\widetilde{\bm{\kappa}}_{b}-\widetilde{\bm{\tau}}_{ab}-\widetilde{\epsilon}\,\widetilde{\bm{g}}_{ab}

\displaystyle=0\,,

(B.1)

where $\widetilde{\bm{\tau}}_{ab}=\widetilde{\bm{\tau}}_{[ab]}=\widetilde{\nabla}_{[a}\widetilde{\bm{\kappa}}_{b]}$ and $\widetilde{\epsilon}=\frac{1}{n+2}\widetilde{\nabla}_{c}\widetilde{k}^{c}$ . In the course of the proof, we shall make use of the additional assumptions


	$\displaystyle\frac{1}{n^{2}}(\widetilde{\nabla}_{a}\widetilde{k}^{a})^{2}-\widetilde{\mathsf{P}}_{ab}\widetilde{k}^{a}\widetilde{k}^{b}-\frac{1}{n}\widetilde{k}^{a}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\widetilde{k}^{b}<0\,,$	(B.2a)
	$\displaystyle\widetilde{k}^{a}\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{d}=2s\widetilde{\bm{\kappa}}_{b}\widetilde{\bm{\kappa}}_{c}\\|\widetilde{\mathsf{W}}(\widetilde{k})\\|^{2}\,,$	(B.2b)
	$\displaystyle\widetilde{k}^{a}\widetilde{\mathsf{Y}}_{abc}\widetilde{k}^{c}=s\widetilde{\bm{\kappa}}_{b}\widetilde{k}^{c}\widetilde{\nabla}_{c}\\|\widetilde{\mathsf{W}}(\widetilde{k})\\|^{2}\,,$	(B.2c)
and
	$\widetilde{\mathsf{W}}_{ab}{}^{cd}\widetilde{\bm{\tau}}_{cd}-2\widetilde{k}^{c}\widetilde{\mathsf{Y}}_{cab}-\frac{1}{2}\left(\widetilde{\bm{\tau}}_{c[a}\widetilde{k}^{d}\widetilde{\mathsf{W}}_{b]d}{}^{ef}\widetilde{\mathsf{W}}_{efg}{}^{c}\widetilde{k}^{g}-\widetilde{\bm{\kappa}}_{[a}\widetilde{k}^{c}\widetilde{\mathsf{W}}_{b]c}{}^{de}\widetilde{\mathsf{Y}}_{fde}\widetilde{k}^{f}\right)\\ =t\widetilde{\nabla}_{[a}\left(\widetilde{\bm{\kappa}}_{b]}\\|\widetilde{\mathsf{W}}(\widetilde{k})\\|^{2}\right)\,,$	(B.2d)

for some real constants $s$ and $t$ , and where $\|\widetilde{\mathsf{W}}(\widetilde{k})\|^{2}:=\widetilde{k}^{a}\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{e}\widetilde{\mathsf{W}}_{e}{}^{bcd}$ . Conditions (B.2a), (B.2b), (B.2c) and (B.2d) are none other than the respective hypotheses (4.2a), (4.2b), (4.2c) and (4.2d) of Theorem 4.6 except that the former depend on two parameters, $s$ and $t$ , while the latter on only one, $\alpha$ . This is for convenience, and an algebraic relation between $s$ , $t$ and $\alpha$ will emerge in the course of the proof.

Let us first assume that equation (B.1) holds. We can always choose a conformal scale $\widetilde{\sigma}$ such that $\widetilde{k}$ is Killing with respect to $\widetilde{g}_{ab}=\sigma^{-2}\widetilde{\bm{g}}_{ab}$ , , i.e. $\mathsterling_{\widetilde{k}}\widetilde{g}_{ab}=0$ . In this case, the prolongation of equation (B.1) then reduces to:


	$\displaystyle\widetilde{\nabla}_{a}\widetilde{\kappa}_{b}-\widetilde{\tau}_{ab}$	$\displaystyle=0\,,$	(B.3a)
	$\displaystyle\widetilde{\nabla}_{a}\widetilde{\tau}_{bc}-2\,\widetilde{g}_{a[b}\widetilde{\psi}_{c]}+2\,\widetilde{\mathsf{P}}{{}_{a[b}}\widetilde{\kappa}_{c]}-\widetilde{k}^{d}\widetilde{\mathsf{W}}_{dabc}$	$\displaystyle=0\,,$	(B.3b)
	$\displaystyle\widetilde{\mathsf{P}}_{a}{}^{b}\widetilde{\kappa}_{b}+\widetilde{\psi}_{a}$	$\displaystyle=0\,,$	(B.3c)
	$\displaystyle\widetilde{\nabla}_{a}\widetilde{\psi}_{b}-\widetilde{\mathsf{P}}_{a}{}^{c}\widetilde{\tau}_{cb}-\widetilde{\mathsf{Y}}_{abc}\widetilde{k}^{c}$	$\displaystyle=0\,,$
where $\widetilde{\kappa}_{a}=\widetilde{g}_{ab}\widetilde{k}^{b}$ , $\widetilde{\tau}_{ab}=\widetilde{\nabla}_{[a}\widetilde{\kappa}_{b]}$ and $\widetilde{\psi}_{a}$ is simply defined by (B.3c). The last display upon skew-symmetrisation becomes

	$\displaystyle\widetilde{\nabla}_{[a}\widetilde{\psi}_{b]}-\widetilde{\mathsf{P}}_{[a}{}^{c}\widetilde{\tau}_{b]c}+\frac{1}{2}\widetilde{k}^{c}\widetilde{\mathsf{Y}}_{cab}$	$\displaystyle=0\,.$	(B.3d)

We shall show that $\widetilde{c}:=\widetilde{\psi}_{c}\widetilde{k}^{c}$ is constant, and that $\widetilde{\tau}_{ab}$ is annihilated by both $\widetilde{k}^{a}$ and $\widetilde{\psi}^{a}$ , and that $\widetilde{\psi}^{a}$ is null. In this way, we will be able to rescale $\widetilde{k}^{a}$ by some constant, so that $\widetilde{\tau}_{ab}$ will play the rôle of a bundle Hermitian structure on the screen bundle of $\widetilde{k}$ . Contracting (B.3a) with $\widetilde{k}^{a}$ already gives

\displaystyle\widetilde{\tau}_{ab}\widetilde{k}^{b}

\displaystyle=0\,,

(B.4)

which also implies that $\widetilde{k}^{b}\widetilde{\nabla}_{b}\widetilde{k}^{a}=0$ . Contracting (B.3b) with $\widetilde{k}^{d}$ , and using (B.3a) and (B.3c) yields

\displaystyle\widetilde{k}^{a}\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{d}

\displaystyle=\widetilde{{\tau}}_{a}{}^{c}\widetilde{{\tau}}_{cb}-(\widetilde{\psi}_{c}\widetilde{k}^{c})\,\widetilde{{g}}_{ab}+2\,\widetilde{{\kappa}}_{(a}\widetilde{\psi}_{b)}\,,

(B.5)

Equation (B.3c) and tracing (B.5) give, with $\widetilde{c}=\widetilde{k}^{a}\widetilde{\psi}_{a}$ ,

	$\displaystyle\widetilde{c}$	$\displaystyle=-\widetilde{\mathsf{P}}_{ab}\widetilde{k}^{a}\widetilde{k}^{b}\,,$		(B.6)
	$\displaystyle\widetilde{c}$	$\displaystyle=-\frac{1}{n}\widetilde{{\tau}}_{ab}\widetilde{{\tau}}^{ab}\,,$		(B.7)

respectively, which upon differentiation, yield

	$\displaystyle\widetilde{\nabla}_{a}\widetilde{c}$	$\displaystyle=-2\widetilde{k}^{b}\widetilde{\mathsf{Y}}_{bac}\widetilde{k}^{c}+\widetilde{{\tau}}_{ac}\widetilde{\psi}^{c}\,,$		(B.8)
	$\displaystyle\widetilde{\nabla}_{a}\widetilde{c}$	$\displaystyle=-\frac{1}{n}\left(4\widetilde{{\tau}}_{ab}\widetilde{\psi}^{b}+2\widetilde{k}^{d}\widetilde{\mathsf{W}}_{dabc}\widetilde{{\tau}}^{bc}\right)\,.$		(B.9)

By the Leibniz rule, equation (B.3a) and the property of $\widetilde{\mathsf{Y}}_{abc}$ , we have

\displaystyle\widetilde{\nabla}^{b}(\widetilde{k}^{a}\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{d})

\displaystyle=-(n-1)\widetilde{k}^{a}\widetilde{\mathsf{Y}}_{acd}\widetilde{k}^{d}+\frac{3}{2}\widetilde{k}^{d}\widetilde{\mathsf{W}}_{dcab}\widetilde{\tau}^{ab}\,.

(B.10)

Henceforth, we impose the assumptions (B.2b) and (B.2c). Recall that the integrability condition for $\widetilde{k}$ to be conformal Killing is that $\mathsterling_{\widetilde{k}}\widetilde{\mathsf{W}}_{abc}{}^{d}=0$ . This means that with our choice of scale for which $\widetilde{k}^{a}$ is Killing, we have that $\mathsterling_{\widetilde{k}}\|\widetilde{\mathsf{W}}(\widetilde{k})\|^{2}=\widetilde{k}^{a}\widetilde{\nabla}_{a}\|\widetilde{\mathsf{W}}(\widetilde{k})\|^{2}=0$ . It immediately follows by (B.2c) that $\widetilde{k}^{a}\widetilde{k}^{c}\widetilde{\mathsf{Y}}_{abc}=0$ . In addition, since $\widetilde{\nabla}^{a}\widetilde{\kappa}_{a}=0$ by (B.3a), and substituting (B.2b) into the LHS of (B.10), it follows that the LHS of (B.10) is identically zero. Hence, we find that (B.10) implies

\displaystyle\widetilde{k}^{d}\widetilde{\mathsf{W}}_{dabc}\widetilde{{\tau}}^{bc}

\displaystyle=0\,.

Hence, (B.8) and (B.9) immediately gives $\widetilde{\nabla}_{a}\widetilde{c}=\widetilde{{\tau}}_{ac}\widetilde{\psi}^{c}=-\frac{4}{n}\widetilde{{\tau}}_{ab}\widetilde{\psi}^{b}$ , i.e. $\widetilde{\nabla}_{a}\widetilde{c}=0$ and $\widetilde{{\tau}}_{ab}\widetilde{\psi}^{b}=0$ . In particular, $\widetilde{c}$ is a constant. Now assuming (B.2a), we conclude that $\widetilde{c}<0$ . This allows us to rescale $\widetilde{k}^{a}$ by $-\frac{1}{\widetilde{c}}$ so that $\widetilde{\psi}_{a}\widetilde{k}^{a}=-1$ . Then, by assumption (4.2b) and equation (B.5), we get

\displaystyle\widetilde{{\tau}}_{a}{}^{c}\widetilde{{\tau}}_{cb}

\displaystyle=-\widetilde{{g}}_{ab}+2\,\widetilde{{\kappa}}_{(a}\widetilde{\lambda}_{b)}\,,

(B.11)

where we have set $\widetilde{\lambda}_{a}=-\widetilde{\psi}_{a}+s\|\widetilde{\mathsf{W}}(\widetilde{k})\|^{2}\widetilde{\kappa}_{a}$ . We also define $\widetilde{\ell}^{a}=\widetilde{g}^{ab}\widetilde{\lambda}_{b}$ .

Since $\widetilde{\ell}^{a}\widetilde{{\tau}}_{ab}=0$ , we immediately conclude that $\widetilde{\ell}^{a}$ is null. Hence, (B.11) tells us that $\widetilde{{\tau}}_{ab}$ defines a bundle Hermitian structure on the screen bundle $\widetilde{H}=\langle\widetilde{k}\rangle^{\perp}/\langle\widetilde{k}\rangle$ of $\langle\widetilde{k}\rangle$ , which is isomorphic to $\langle\widetilde{k}\rangle^{\perp}\cap\langle\widetilde{\ell}\rangle^{\perp}$ . By virtue of the fact that $\widetilde{k}$ is null and conformal Killing, $\widetilde{k}$ generates a non-shearing congruence of null geodesics $\widetilde{\mathcal{K}}$ . Furthermore, the algebraic condition (B.11) tells us that the twist of $\widetilde{\mathcal{K}}$ , identified with $\widetilde{{\tau}}_{ab}$ , induces an almost Robinson structure $(\widetilde{N},\langle\widetilde{k}\rangle)$ on $\widetilde{\mathcal{M}}$ . Hence, by [FLTC23, TC22], $(\widetilde{N},\langle\widetilde{k}\rangle)$ induces a partially integrable almost contact CR structure $(H,J)$ on the local leaf space $\widetilde{\mathcal{M}}$ of $\widetilde{\mathcal{K}}$ . Our choice of metric $\widetilde{g}$ for which $\widetilde{k}$ is Killing corresponds to the pseudo-hermitian form $\theta=\frac{1}{2}\widetilde{\kappa}$ on $\mathcal{M}$ with Levi form $h=\widetilde{\tau}$ .

Let us now impose the condition (B.2d). Plugging $\widetilde{\mathsf{R}}_{ab}{}^{cd}\widetilde{\tau}_{bc}=\widetilde{\mathsf{W}}_{ab}{}^{cd}\widetilde{\tau}_{bc}-4\widetilde{\mathsf{P}}_{[a}{}^{c}\widetilde{\tau}_{b]c}$ into (B.3d) and using (B.2d) yields

\widetilde{\nabla}_{[a}\widetilde{\psi}_{b]}=-\frac{1}{4}\widetilde{R}_{ab}{}^{cd}\widetilde{\tau}_{cd}+\frac{1}{4}\left(\frac{1}{2}\left(\widetilde{{\tau}}_{c[a}\widetilde{k}^{d}\widetilde{\mathsf{W}}_{b]d}{}^{ef}\widetilde{\mathsf{W}}_{efg}{}^{c}\widetilde{k}^{g}-\widetilde{{\kappa}}_{[a}\widetilde{k}^{c}\widetilde{\mathsf{W}}_{b]c}{}^{de}\widetilde{\mathsf{Y}}_{fde}\widetilde{k}^{f}\right)\right)\\ +\frac{1}{4}t\widetilde{\nabla}_{[a}\left(\widetilde{{\kappa}}_{b]}\|\widetilde{\mathsf{W}}(\widetilde{k})\|^{2}\right)\,.

(B.12)

In addition, (B.3d) gives

\displaystyle\widetilde{{\kappa}}_{[a}\widetilde{k}^{c}\widetilde{\mathsf{W}}_{b]c}{}^{de}\widetilde{\mathsf{Y}}_{fde}\widetilde{k}^{f}

\displaystyle=2\,\widetilde{{\kappa}}_{[a}\widetilde{k}^{c}\widetilde{\mathsf{W}}_{b]c}{}^{de}\left(\widetilde{\nabla}_{d}\widetilde{\psi}_{e}+\widetilde{\mathsf{P}}_{d}{}^{f}\widetilde{{\tau}}_{ef}\right)\,.

(B.13)

Since $\widetilde{k}$ generates a twisting non-shearing congruence of null geodesics, non-expanding with respect to the chosen $\widetilde{g}$ , we may use the computation of the curvature given in Appendix A of [TC22]. It is a tedious computational matter to show that $\widetilde{{\kappa}}_{[a}\widetilde{k}^{c}\widetilde{\mathsf{W}}_{b]c}{}^{de}\widetilde{\mathsf{P}}_{d}{}^{f}\widetilde{{\tau}}_{ef}=0$ .

At this stage, let us write

\displaystyle\widetilde{\nabla}_{[a}\widetilde{\lambda}_{b]}=-\widetilde{B}_{ab}+\widetilde{{\kappa}}_{[a}\widetilde{C}_{b]}\,,

(B.14)

for some tensors $\widetilde{B}_{ab}=\widetilde{B}_{[ab]}$ and $\widetilde{C}_{a}$ . Since $\widetilde{k}$ is conformal Killing, we have $\widetilde{k}^{a}\widetilde{B}_{ab}=\widetilde{k}^{a}\widetilde{C}_{a}=0$ and $\widetilde{\ell}^{a}\widetilde{B}_{ab}=\widetilde{\ell}^{a}\widetilde{C}_{a}=0$ . Using $\widetilde{\psi}_{a}=-\widetilde{\lambda}_{a}+s\|\widetilde{\mathsf{W}}(\widetilde{k})\|^{2}\widetilde{\kappa}_{a}$ , combining the previous display with (B.13), we obtain

\widetilde{B}_{ab}-\widetilde{{\kappa}}_{[a}\widetilde{C}_{b]}=-\frac{1}{4}\widetilde{\mathsf{R}}_{ab}{}^{cd}\widetilde{\tau}_{cd}+\frac{1}{8}\left(\widetilde{{\tau}}_{c[a}\widetilde{k}^{d}\widetilde{\mathsf{W}}_{b]d}{}^{ef}\widetilde{\mathsf{W}}_{efg}{}^{c}\widetilde{k}^{g}-2\,\widetilde{{\kappa}}_{[a}\widetilde{k}^{c}\widetilde{\mathsf{W}}_{b]c}{}^{de}\widetilde{B}_{de}\right)\\ +\frac{1}{4}(t-4s)\left(\|\widetilde{\mathsf{W}}(\widetilde{k})\|^{2}\widetilde{\tau}_{ab}-\widetilde{{\kappa}}_{[a}\widetilde{\nabla}_{b]}\|\widetilde{\mathsf{W}}(\widetilde{k})\|^{2}\right)\,.

(B.15)

We take the components of $\widetilde{B}_{ab}$ with respect to the splitting of $\langle\widetilde{k}\rangle^{\perp}\cap\langle\widetilde{\ell}\rangle^{\perp}$ into the eigenbundles of the bundle complex structure defined by $\widetilde{\tau}_{ab}$ . In the obvious index notation, we find

$\displaystyle\widetilde{B}_{\alpha\beta}$	$\displaystyle=-\frac{1}{4}\widetilde{\mathsf{R}}_{\alpha\beta}{}^{cd}\widetilde{\tau}_{cd}\,,$	(B.16)
$\displaystyle\widetilde{B}_{\alpha\bar{\beta}}$	$\displaystyle=-\frac{1}{4}\widetilde{\mathsf{R}}_{\alpha\bar{\beta}}{}^{cd}\widetilde{\tau}_{cd}-\frac{1}{8}\mathrm{i}\widetilde{k}^{d}\widetilde{\mathsf{W}}_{\alpha d}{}^{ef}\widetilde{\mathsf{W}}_{efg\bar{\beta}}\widetilde{k}^{g}+\frac{1}{4}(t-4s)\mathrm{i}\\|\widetilde{\mathsf{W}}(\widetilde{k})\\|^{2}h_{\alpha\bar{\beta}}\,,$	(B.17)
$\displaystyle\widetilde{C}_{\alpha}$	$\displaystyle=-\frac{1}{2}\widetilde{\mathsf{R}}_{\alpha 0\bar{\beta}}{}^{cd}\widetilde{\tau}_{cd}+\frac{1}{4}\left(-\frac{\mathrm{i}}{2}\widetilde{k}^{d}\widetilde{\mathsf{W}}_{0d}{}^{ef}\widetilde{\mathsf{W}}_{efg\alpha}\widetilde{k}^{g}+\widetilde{k}^{c}\widetilde{\mathsf{W}}_{\alpha c}{}^{de}\widetilde{B}_{de}\right)+\frac{1}{4}(t-4s)\widetilde{\nabla}_{\alpha}\\|\widetilde{\mathsf{W}}(\widetilde{k})\\|^{2}\,.$	(B.18)

Referring again to the computation of the curvature given in [TC22], tracing equations (A.10), (A.11) and (A.14) over the first two indices yields

$\displaystyle\widetilde{\mathsf{R}}_{\gamma}{}^{\gamma}{}_{\alpha\beta}$	$\displaystyle=2(m+1)\,\mathrm{i}\widetilde{B}_{\alpha\beta}-\nabla^{\gamma}\mathsf{N}_{\alpha\beta\gamma}\,,$	(B.19)
$\displaystyle\widetilde{\mathsf{R}}_{\gamma}{}^{\gamma}{}_{\bar{\beta}\alpha}$	$\displaystyle=\mathsf{Ric}_{\alpha\bar{\beta}}-2(m+1)\,\mathrm{i}\widetilde{B}_{\alpha\bar{\beta}}-2\,\mathrm{i}\widetilde{B}_{\gamma}{}^{\gamma}h_{\alpha\bar{\beta}}+\mathsf{N}_{\alpha\epsilon\gamma}\mathsf{N}{}_{\bar{\beta}}{}^{\epsilon\gamma}\,,$	(B.20)
$\displaystyle\widetilde{\mathsf{R}}_{\beta}{}^{\beta}{}_{\alpha 0}$	$\displaystyle=-\nabla^{\beta}\widetilde{B}_{\alpha\beta}+\nabla_{\beta}\widetilde{B}_{\alpha}{}^{\beta}+2m\,\mathrm{i}\widetilde{C}_{\alpha}+\widetilde{B}^{\beta\delta}\mathsf{N}_{\delta\alpha\beta}-\frac{1}{2}\nabla^{\beta}\mathsf{A}_{\alpha\beta}-\frac{1}{2}\mathsf{A}^{\beta\delta}\mathsf{N}_{\delta\alpha\beta}\,.$	(B.21)

Here, $\nabla$ is the Webster connection corresponding to $\theta$ , $\mathsf{Ric}_{\alpha\bar{\beta}}$ its Webster–Ricci tensor and $\mathsf{N}_{\alpha\beta\gamma}$ the Nijenhuis tensor. Similarly, we compute

$\displaystyle\widetilde{k}^{d}\widetilde{\mathsf{W}}_{\beta d}{}^{ef}\widetilde{\mathsf{W}}_{efg}{}^{\alpha}\widetilde{k}^{g}$	$\displaystyle=-4\mathsf{N}_{\gamma\delta\beta}\mathsf{N}^{\gamma\delta\alpha}\,,$	(B.22)
$\displaystyle\widetilde{k}^{d}\widetilde{\mathsf{W}}_{0d}{}^{ef}\widetilde{\mathsf{W}}_{efg\alpha}\widetilde{k}^{g}$	$\displaystyle=-4\mathsf{N}_{\gamma\delta\alpha}\widetilde{B}^{\gamma\delta}\,,$	(B.23)
$\displaystyle\widetilde{k}^{c}\widetilde{\mathsf{W}}_{\beta c}{}^{de}\widetilde{B}_{de}$	$\displaystyle=2\mathrm{i}\mathsf{N}_{\gamma\delta\beta}\widetilde{B}^{\gamma\delta}\,.$	(B.24)

This means that

\displaystyle\|\widetilde{\mathsf{W}}(\widetilde{k})\|^{2}

\displaystyle:=\widetilde{k}^{a}\widetilde{\mathsf{W}}_{abcd}\widetilde{k}^{e}\widetilde{\mathsf{W}}_{e}{}^{bcd}=8\|\mathsf{N}\|^{2}\,.

(B.25)

We now note $\widetilde{\mathsf{R}}_{ab}{}^{cd}\widetilde{\tau}_{bc}=-2\mathrm{i}\widetilde{\mathsf{R}}_{ab\gamma}{}^{\gamma}$ . Hence, plugging (B.19) into (B.16) and solving for $B_{\alpha\beta}$ gives

\displaystyle\widetilde{B}_{\alpha\beta}

\displaystyle=-\frac{1}{2(m+2)}\mathrm{i}\nabla^{\gamma}\mathsf{N}_{\alpha\beta\gamma}\,.

(B.26)

Similarly, plugging (B.20) and (B.22) into (B.17) and solving for $\widetilde{B}_{\alpha\bar{\beta}}$ leads to

\displaystyle\widetilde{B}_{\alpha\bar{\beta}}

\displaystyle=-\frac{1}{2(m+2)}\mathrm{i}\left(\mathsf{Ric}_{\alpha\bar{\beta}}-\mathsf{N}_{\gamma\delta\alpha}\mathsf{N}^{\gamma\delta}{}_{\bar{\beta}}+\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}\right)-\frac{1}{m+2}\widetilde{B}_{\gamma}{}^{\gamma}h_{\alpha\bar{\beta}}+2(t-4s)\mathrm{i}\|\mathsf{N}\|^{2}h_{\alpha\bar{\beta}}\,.

Taking the trace and solving for $B_{\gamma}{}^{\gamma}$ gives

\displaystyle\widetilde{B}_{\gamma}{}^{\gamma}

\displaystyle=-\frac{1}{2}\mathrm{i}\mathsf{P}+(t-4s)\frac{m(m+2)}{m+1}\mathrm{i}\|\mathsf{N}\|^{2}\,.

By substituting back into the last but one equation, we obtain

\displaystyle\widetilde{B}_{\alpha\bar{\beta}}

\displaystyle=-\frac{1}{2(m+2)}\mathrm{i}\left(\mathsf{Ric}_{\alpha\bar{\beta}}-\mathsf{N}_{\gamma\delta\alpha}\mathsf{N}^{\gamma\delta}{}_{\bar{\beta}}+\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}-\mathsf{P}h_{\alpha\bar{\beta}}\right)+(t-4s)\frac{m+2}{m+1}\mathrm{i}\|\mathsf{N}\|^{2}h_{\alpha\bar{\beta}}\,.

On the other hand, condition (B.2b) also gives us, with reference to Appendix A of [TC22],

\displaystyle 16s\|\mathsf{N}\|^{2}

\displaystyle=\frac{1}{2m+1}\left(1-4m(m+2)(t-4s)\right)\|\mathsf{N}\|^{2}\,.

Assuming that $\|\mathsf{N}\|^{2}$ nowhere vanishes, this implies

\displaystyle t-4s

\displaystyle=\frac{1-16(2m+1)s}{4m(m+2)}\,.

(B.27)

Thus,

\displaystyle\widetilde{B}_{\alpha\bar{\beta}}

\displaystyle=-\frac{1}{2(m+2)}\mathrm{i}\left(\mathsf{Ric}_{\alpha\bar{\beta}}-\mathsf{N}_{\gamma\delta\alpha}\mathsf{N}^{\gamma\delta}{}_{\bar{\beta}}+\mathsf{N}_{\alpha\gamma\delta}\mathsf{N}_{\bar{\beta}}{}^{\gamma\delta}-\mathsf{P}h_{\alpha\bar{\beta}}\right)+\frac{1-16(2m+1)s}{4m(m+1)}\mathrm{i}\|\mathsf{N}\|^{2}h_{\alpha\bar{\beta}}\,.

(B.28)

Finally, by plugging (B.21), (B.23) and (B.24) into (B.18) and solving for $\widetilde{C}_{\alpha}$ , we find

\displaystyle\widetilde{C}_{\alpha}

\displaystyle=\frac{1}{2}T_{\alpha}+\frac{1-16(2m+1)s}{4m(m+1)}\nabla_{\alpha}\|\mathsf{N}\|^{2}\,,

(B.29)

where $T_{\alpha}$ is given by (2.6).

Let $(\widetilde{\mathcal{M}}^{\prime},\widetilde{\mathbf{c}}^{\prime})$ be the $\alpha$ -Fefferman space of $(\mathcal{M},H,J)$ . Let $\widetilde{\omega}_{\theta}$ be the induced Webster connection on $\widetilde{\mathcal{M}}$ compatible with the contact form $\theta$ , and denote $\widetilde{g}^{\prime}$ its corresponding Fefferman metric. We set

\displaystyle\widetilde{\lambda}^{\prime}

\displaystyle=\widetilde{\omega}_{\theta}-\frac{1}{m+2}\left(\mathsf{P}+\frac{\alpha(m+2)}{2m(m+1)}\|\mathsf{N}\|^{2}\right)\theta\,,

so that $\widetilde{g}^{\prime}=4\theta\odot\widetilde{\lambda}^{\prime}+h$ . We find that

\displaystyle\mathrm{d}\widetilde{\lambda}^{\prime}

\displaystyle=-\widetilde{B}^{\prime}_{\alpha\beta}\theta^{\alpha}\wedge\theta^{\beta}-2\widetilde{B}^{\prime}_{\alpha\bar{\beta}}\theta^{\alpha}\wedge\overline{\theta}{}^{\bar{\beta}}-\widetilde{B}^{\prime}_{\bar{\alpha}\bar{\beta}}\overline{\theta}{}^{\bar{\alpha}}\wedge\overline{\theta}{}^{\bar{\beta}}-2\widetilde{C}^{\prime}_{\bar{\alpha}}\overline{\theta}{}^{\bar{\alpha}}\wedge\theta-2\widetilde{C}^{\prime}_{\alpha}\theta^{\alpha}\wedge\theta\,,

where


$\displaystyle\widetilde{B}^{\prime}_{\alpha\bar{\beta}}$	$\displaystyle=-\frac{\mathrm{i}}{2}{\mathsf{P}}_{\alpha\bar{\beta}}-\frac{\mathrm{i}}{2(m+2)}\left({\mathsf{N}}_{\alpha\gamma\delta}{\mathsf{N}}_{\bar{\beta}}{}^{\gamma\delta}-{\mathsf{N}}_{\gamma\delta\alpha}{\mathsf{N}}^{\gamma\delta}{}_{\bar{\beta}}\right)+\alpha\frac{\mathrm{i}}{4m(m+1)}\\|\mathsf{N}\\|^{2}h_{\alpha\bar{\beta}}\,,$	(B.30a)
$\displaystyle\widetilde{B}^{\prime}_{\gamma\delta}$	$\displaystyle=-\frac{\mathrm{i}}{2(m+2)}{\nabla}^{\alpha}{\mathsf{N}}_{\gamma\delta\alpha}\,,$	(B.30b)
$\displaystyle\widetilde{C}^{\prime}_{\alpha}$	$\displaystyle=\frac{1}{2}{T}_{\alpha}+\alpha\frac{1}{4m(m+1)}\nabla_{\alpha}\left(\\|\mathsf{N}\\|^{2}\right)\,,$	(B.30c)

Hence, setting

\displaystyle s

\displaystyle=\frac{1-\alpha}{16(2m+1)}\,,

\displaystyle t

\displaystyle=\frac{1}{4(2m+1)}\left(1-\frac{m^{2}-1}{m(m+2)}\alpha\right)\,,

so that $\alpha=1-16(2m+1)s$ , we see that conditions (B.2a), (B.2b), (B.2c) and (B.2d) are none other than the respective hypotheses (4.2a), (4.2b), (4.2c) and (4.2d) of Theorem 4.6. In addition, on comparing (B.28), (B.26) and (B.29) with (B.30a), (B.30b) and (B.30c), we find $\mathrm{d}\widetilde{\lambda}=\mathrm{d}\widetilde{\lambda}^{\prime}$ , which locally implies that

\displaystyle\widetilde{\lambda}^{\prime}-\widetilde{\lambda}

\displaystyle=\mathrm{d}\phi^{\prime}\,.

for some smooth function $\phi^{\prime}$ on $\widetilde{\mathcal{M}}^{\prime}$ . Choose some density $\sigma$ with corresponding local coordinate $\phi$ , and define a map $\iota$ from $(\widetilde{\mathcal{M}}^{\prime},\widetilde{g}^{\prime})$ to $(\widetilde{\mathcal{M}},\widetilde{g})$ by $\iota(p)=(\varpi^{\prime}(p),\mathrm{e}^{\mathrm{i}\phi^{\prime}(p)}\sigma)$ for any $p\in\widetilde{\mathcal{M}}^{\prime}$ . Then $\iota$ is a bundle map that is also a local isometry. It now follows that $(\widetilde{\mathcal{M}}^{\prime},\widetilde{\mathbf{c}}^{\prime})$ and $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}})$ are locally conformally isometric as claimed.

$\displaystyle\widetilde{g}_{\theta}$

$\displaystyle=4\theta\odot\widetilde{\lambda}+h\,,$

where $h$ is (the metric induced from) the Levi form of $\theta$ , and $\widetilde{\lambda}$ is a unique null one-form satisfying $\widetilde{k}\makebox[7.0pt]{\rule{6.0pt}{0.3pt}\rule{0.3pt}{5.0pt}}\,\widetilde{\lambda}=1$ . Choosing an admissible coframe $(\theta^{\alpha})$ for $\theta$ , and an affine parameter $\phi$ along the geodesics of $\widetilde{k}$ so that $\widetilde{k}=\frac{\partial}{\partial\phi}$ , we can write

$\displaystyle h$

$\displaystyle=2h_{\alpha\bar{\beta}}\theta^{\alpha}\odot\overline{\theta}{}^{\bar{\beta}}\,,$

$\displaystyle\widetilde{\lambda}$

$\displaystyle=\mathrm{d}\phi+\widetilde{\lambda}_{\alpha}\theta^{\alpha}+\widetilde{\lambda}_{\bar{\alpha}}\overline{\theta}{}^{\bar{\alpha}}+\widetilde{\lambda}_{0}\theta^{0}\,,$

[BEG94] T. N. Bailey, M. G. Eastwood and A. R. Gover “Thomas’s structure bundle for conformal, projective and related structures” In Rocky Mountain J. Math. 24.4, 1994, pp. 1191–1217
[Bau] Helga Baum “Holonomy of conformal structures” Notes from lectures given at the seminar ‘Recent Developments in Conformal Differential Geometry’, Mathematisches Forschungsinstitut Oberwolfach seminar, 18 ? 24 November 2007.
[ČG08] Andreas Čap and A. Rod Gover “CR-tractors and the Fefferman space” In Indiana Univ. Math. J. 57.5, 2008, pp. 2519–2570 DOI: 10.1512/iumj.2008.57.3359
[ČG10] Andreas Čap and A. Rod Gover “A holonomy characterisation of Fefferman spaces” In Ann. Global Anal. Geom. 38.4, 2010, pp. 399–412 DOI: 10.1007/s10455-010-9220-6
[CG18] Sean N. Curry and A. Rod Gover “An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity” In Asymptotic analysis in general relativity 443, London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 2018, pp. 86–170
[CG20] Jeffrey S. Case and A. Rod Gover “The $P^{\prime}$ -operator, the $Q^{\prime}$ -curvature, and the CR tractor calculus” In Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20.2, 2020, pp. 565–618 arXiv:1709.08057
[FG85] Charles Fefferman and C. Robin Graham “Conformal invariants” The mathematical heritage of Élie Cartan (Lyon, 1984) In Astérisque, 1985, pp. 95–116
[FLTC20] Anna Fino, Thomas Leistner and Arman Taghavi-Chabert “Optical geometries” Accepted for publication in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2020 arXiv:2009.10012 [math.DG]
[FLTC23] Anna Fino, Thomas Leistner and Arman Taghavi-Chabert “Almost Robinson geometries” In Lett. Math. Phys. 113.3, 2023, pp. 56 DOI: 10.1007/s11005-023-01667-x
[Fef76] Charles L. Fefferman “Correction to: “Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains” (Ann. of Math. (2) 103 (1976), no. 2, 395–416)” In Ann. of Math. (2) 104.2, 1976, pp. 393–394 DOI: 10.2307/1970961
[Fef76a] Charles L. Fefferman “Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains” In Ann. of Math. (2) 103.2, 1976, pp. 395–416 DOI: 10.2307/1970945
[GG05] A. Rod Gover and C. Robin Graham “CR invariant powers of the sub-Laplacian” In J. Reine Angew. Math. 583, 2005, pp. 1–27 DOI: 10.1515/crll.2005.2005.583.1
[GP06] A. Rod Gover and Lawrence J. Peterson “The ambient obstruction tensor and the conformal deformation complex” In Pacific J. Math. 226.2, 2006, pp. 309–351 DOI: 10.2140/pjm.2006.226.309
[Gov05] A. Rod Gover “Almost conformally Einstein manifolds and obstructions” In Differential geometry and its applications Matfyzpress, Prague, 2005, pp. 247–260
[Gra87] C. Robin Graham “On Sparling’s characterization of Fefferman metrics” In Amer. J. Math. 109.5, 1987, pp. 853–874 DOI: 10.2307/2374491
[HLN08] C. Denson Hill, Jerzy Lewandowski and Paweł Nurowski “Einstein’s equations and the embedding of 3-dimensional CR manifolds” In Indiana Univ. Math. J. 57.7, 2008, pp. 3131–3176 DOI: 10.1512/iumj.2008.57.3473
[Hau74] I. Hauser “Type- $N$ gravitational field with twist” In Phys. Rev. Lett. 33.18, 1974, pp. 1112–1113; erratum, ibid. 33 (1974), no. 25, 1525 DOI: 10.1103/PhysRevLett.33.1112
[Ker63] Roy P. Kerr “Gravitational field of a spinning mass as an example of algebraically special metrics” In Phys. Rev. Lett. 11, 1963, pp. 237–238
[LN90] Jerzy Lewandowski and Paweł Nurowski “Algebraically special twisting gravitational fields and CR structures” In Classical Quantum Gravity 7.3, 1990, pp. 309–328 URL: http://stacks.iop.org/0264-9381/7/309
[LNT91] Jerzy Lewandowski, Paweł Nurowski and Jacek Tafel “Algebraically special solutions of the Einstein equations with pure radiation fields” In Classical Quantum Gravity 8.3, 1991, pp. 493–501 URL: http://stacks.iop.org/0264-9381/8/493
[LeB85] Claude R. LeBrun “Ambi-twistors and Einstein’s equations” In Classical Quantum Gravity 2.4, 1985, pp. 555–563 URL: http://stacks.iop.org/0264-9381/2/555
[Lee86] John M. Lee “The Fefferman metric and pseudo-Hermitian invariants” In Trans. Amer. Math. Soc. 296.1, 1986, pp. 411–429 DOI: 10.2307/2000582
[Lee88] John M. Lee “Pseudo-Einstein structures on CR manifolds” In Amer. J. Math. 110.1, 1988, pp. 157–178 DOI: 10.2307/2374543
[Lei07] Felipe Leitner “On transversally symmetric pseudo-Einstein and Fefferman-Einstein spaces” In Math. Z. 256.2, 2007, pp. 443–459 DOI: 10.1007/s00209-007-0121-8
[Lei10] Felipe Leitner “A gauged Fefferman construction for partially integrable CR geometry” In J. Geom. Phys. 60.9, 2010, pp. 1262–1278 DOI: 10.1016/j.geomphys.2010.03.009
[Lew91] Jerzy Lewandowski “Twistor equation in a curved space-time” In Class. Quant. Grav. 8, 1991, pp. L11–L18
[Mas98] L. J. Mason “The asymptotic structure of algebraically special spacetimes” In Classical Quantum Gravity 15.4, 1998, pp. 1019–1030 DOI: 10.1088/0264-9381/15/4/022
[Mat16] Yoshihiko Matsumoto “GJMS operators, $Q$ -curvature, and obstruction tensor of partially integrable CR manifolds” In Differential Geom. Appl. 45, 2016, pp. 78–114 DOI: 10.1016/j.difgeo.2016.01.002
[Mat22] Yoshihiko Matsumoto “The CR Killing operator and Bernstein-Gelfand-Gelfand construction in CR geometry”, 2022 arXiv:2205.11022 [math.DG]
[NP99] Paweł Nurowski and Maciej Przanowski “A four-dimensional example of a Ricci flat metric admitting almost-Kähler non-Kähler structure” In Classical Quantum Gravity 16.3, 1999, pp. L9–L13 DOI: 10.1088/0264-9381/16/3/002
[NT02] P. Nurowski and A. Trautman “Robinson manifolds as the Lorentzian analogs of Hermite manifolds” 8th International Conference on Differential Geometry and its Applications (Opava, 2001) In Differential Geom. Appl. 17.2-3, 2002, pp. 175–195
[NTU63] E. Newman, L. Tamburino and T. Unti “Empty-space generalization of the Schwarzschild metric” In J. Mathematical Phys. 4, 1963, pp. 915–923
[PR86] Roger Penrose and Wolfgang Rindler “Spinors and space-time. Vol. 2” Spinor and twistor methods in space-time geometry, Cambridge Monographs on Mathematical Physics Cambridge: Cambridge University Press, 1986
[Pen65] R. Penrose “Zero rest-mass fields including gravitation: Asymptotic behaviour” In Proc. Roy. Soc. London Ser. A 284, 1965, pp. 159–203 DOI: 10.1098/rspa.1965.0058
[Pen67] Roger Penrose “Twistor algebra” In J. Math. Phys. 8, 1967, pp. 345–366
[RT85] Ivor Robinson and Andrzej Trautman “Integrable optical geometry” In Letters in Mathematical Physics 10, 1985, pp. 179–182
[RT86] Ivor Robinson and Andrzej Trautman “Cauchy-Riemann structures in optical geometry” In Proceedings of the fourth Marcel Grossmann meeting on general relativity, Part A, B (Rome, 1985) North-Holland, Amsterdam, 1986, pp. 317–324
[RT89] Ivor Robinson and Andrzej Trautman “Optical geometry” In New Theories in Physics, Proc. of the XI Warsaw Symp. on Elementary Particle Physics, Kazimierz 23-27 May 1988 World Scientific, 1989, pp. 454–497
[TC22] Arman Taghavi-Chabert “Twisting non-shearing congruences of null geodesics, almost CR structures and Einstein metrics in even dimensions” In Ann. Mat. Pura Appl. (4) 201.2, 2022, pp. 655–693 DOI: 10.1007/s10231-021-01133-2
[TC23] Arman Taghavi-Chabert “Perturbations of Fefferman spaces over CR three-manifolds”, preprint arXiv:2303.07328, 2023 arXiv:2303.07328 [math.DG]
[Taf85] Jacek Tafel “On the Robinson theorem and shearfree geodesic null congruences” In Lett. Math. Phys. 10.1, 1985, pp. 33–39 DOI: 10.1007/BF00704584
[Tan75] Noboru Tanaka “A differential geometric study on strongly pseudo-convex manifolds” Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9 Kinokuniya Book-Store Co., Ltd., Tokyo, 1975, pp. iv+158
[Tau51] A. H. Taub “Empty space-times admitting a three parameter group of motions” In Ann. of Math. (2) 53, 1951, pp. 472–490 DOI: 10.2307/1969567
[Tra02] Andrzej Trautman “Robinson manifolds and Cauchy-Riemann spaces” In Classical Quantum Gravity 19.2, 2002, pp. R1–R10
[Tra84] Andrzej Trautman “Deformations of the Hodge map and optical geometry” In J. Geom. Phys. 1.2, 1984, pp. 85–95 DOI: 10.1016/0393-0440(84)90005-6
[Tra85] Andrzej Trautman “Optical structures in relativistic theories” The mathematical heritage of Élie Cartan (Lyon, 1984) In Astérisque, 1985, pp. 401–420
[Tra99] Andrzej Trautman “Gauge and optical aspects of gravitation” In Classical Quantum Gravity 16.12A, 1999, pp. A157–A175 DOI: 10.1088/0264-9381/16/12A/308
[Web78] S. M. Webster “Pseudo-Hermitian structures on a real hypersurface” In J. Differential Geometry 13.1, 1978, pp. 25–41 URL: http://projecteuclid.org/euclid.jdg/1214434345
[Yau00] S.-T. Yau “Review of geometry and analysis” In Mathematics: frontiers and perspectives Amer. Math. Soc., Providence, RI, 2000, pp. 353–401 DOI: 10.1016/s0378-3758(98)00182-7