On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric

Yoshinobu Kamishima

Department of Mathematics, Josai University, Saitama, Japan.

DOI: 10.4236/am.2018.92008                

Abstract

A CR-structure on a 2n +1-manifold gives a conformal class of Lorentz metrics on the Fefferman S¹-bundle. This analogy is carried out to the quarternionic conformal 3-CR structure (a generalization of quaternionic CR- structure) on a 4n + 3 -manifold M. This structure produces a conformal class [g] of a pseudo-Riemannian metric g of type (4n + 3,3) on M × S³. Let (PSp(n +1,1), S4ⁿ⁺³) be the geometric model obtained from the projective boundary of the complete simply connected quaternionic hyperbolic manifold. We shall prove that M is locally modeled on (PSp(n +1,1), S4ⁿ⁺³) if and only if (M × S³ ,[g]) is conformally flat (i.e. the Weyl conformal curvature tensor vanishes).

Keywords

Conformal Structure, Quaternionic CR-Structure, G-Structure, Conformally Flat Structure, Weyl Tensor, Integrability, Uniformization, Transformation Groups

1. Introduction

This paper concerns a geometric structure on $\left(4n+3\right)$ -manifolds which is related with CR-structure and also quaternionic CR-structure (cf. [1] [2] ). Given a quaternionic CR-structure ${\left\{{\omega }_{\alpha }\right\}}_{\alpha =1,2,3}$ on a $4n+3$ -manifold M, we have proved in [3] that the associated endomorphism $J_{\alpha }$ on the 4n-bundle $\text{D}$ naturally extends to a complex structure ${\bar{J}}_{\alpha }$ on $\text{ker}{\omega }_{\alpha }$ . So we obtain 3 CR-structures on M. Taking into account this fact, we study the following geometric structure on $\left(4n+3\right)$ -manifolds globally.

A hypercomplex 3 CR-structure on a $\left(4n+3\right)$ -manifold M consists of (positive definite) 3 pseudo-Hermitian structures ${\left\{{\omega }_{\alpha },J_{\alpha }\right\}}_{\alpha =1,2,3}$ on M which satisfies that

1) $\text{D}={\displaystyle \underset{\alpha =1}{\overset{3}{\cap }}\text{ker}{\omega }_{\alpha }}$ is a 4n-dimensional subbundle of TM such that

$\text{D}+\left[\text{D}\mathrm{,}\text{D}\right]=TM$ .

2) Each $J_{\gamma }$ coincides with the endomorphism ${\left(d{\omega }_{\beta }\mathrm{<mo>\; |\; </mo>}\text{D}\right)}^{-1}\circ \left(d{\omega }_{\alpha }\mathrm{<mo>\; |\; </mo>}\text{D}\right)\mathrm{<mo>\; :\; </mo>}\text{D}\to \text{D}$ $\left(\left(\alpha \mathrm{,}\beta \mathrm{,}\gamma \right)~\left(\mathrm{1,2,3}\right)\right)$ such that $\left\{J_1\mathrm{,}{J}_2\mathrm{,}{J}_3\right\}$ constitutes a hypercomplex structure on $\text{D}$ .

We call the pair $\left(\text{D}\mathrm{,}\left\{J_1\mathrm{,}{J}_2\mathrm{,}{J}_3\right\}\right)$ also a hypercomplex 3 CR-structure if it is represented by such pseudo-Hermitian structures on M. A quaternionic CR- structure is an example of our hypercomplex 3 CR-structure. As Sasakian 3- structure is equivalent with quaternionic CR-structure, Sasakian 3-structure is also an example. Especially the $4n+3$ -dimensional standard sphere $S^{4n+3}$ is a hypercomplex 3 CR-manifold. The pair $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\mathrm{,}{S}^{4n+3}\right)$ is the spherical homogeneous model of hypercomplex 3 CR-structure in the sense of Cartan geometry (cf. [4] ). First we study the properties of hypercomplex 3 CR-structure. Next we introduce a quaternionic 3 CR-structure on M in a local manner. In fact, let $\text{D}$ be a 4n-dimensional subbundle endowed with a quaternionic structure Q on a $\left(4n+3\right)$ -manifold M. The pair $\left(\text{D}\mathrm{,}Q\right)$ is called quaternionic 3 CR-structure if the following conditions hold:

1) $\text{D}+\left[\text{D}\mathrm{,}\text{D}\right]=TM$ ;

2) M has an open cover ${\left\{U_i\right\}}_{i\in \Lambda }$ each $U_i$ of which admits a hypercomplex 3 CR-structure ${\left({\omega }_{\alpha }^{\left(i\right)},J_{\alpha }^{\left(i\right)}\right)}_{\alpha =1,2,3}$ such that:

a) $\text{D}\mathrm{<mo>\; |\; </mo>}{U}_i={\displaystyle \underset{\alpha =1}{\overset{3}{\cap }}\text{ker}{\omega }_{\alpha }^{\left(i\right)}}$ ;

b) Each hypercomplex structure ${\left\{J_1^{\left(i\right)}\mathrm{,}{J}_2^{\left(i\right)}\mathrm{,}{J}_3^{\left(i\right)}\right\}}_{i\in \Lambda }$ on $\text{D}\mathrm{<mo>\; |\; </mo>}{U}_i$ generates a quaternionic structure Q on $\text{D}$ .

A $4n+3$ -manifold equipped with this structure is said to be a quaternionic 3 CR-manifold. A typical example of a quaternionic 3 CR-manifold but not a hypercomplex 3 CR-manifold is a quaterninic Heisenberg nilmanifold. In this paper, we shall study an invariant for quaternionic 3 CR-structure on $\left(4n+3\right)$ - manifolds.

Theorem A. Let $\left(M\mathrm{,}\left\{\text{D}\mathrm{,}Q\right\}\right)$ be a quaternionic 3 CR-manifold. There exists a pseudo-Riemannian metric g of type $\left(4n+\mathrm{3,3}\right)$ on $M\times S^3$ . Then the con- formal class $\left[g\right]$ is an invariant for quaternionic 3 CR-structure.

As well as the spherical quaternionic 3 CR homogeneous manifold $S^{4n+3}$ , we have the pseudo-Riemannian homogeneous manifold $S^{4n+3}\times S^3$ which is a two-fold covering of the pseudo-Riemannian homogeneous manifold $\left(S^{4n+3}{\times }_{{\mathbb{Z}}_2}{S}^3\mathrm{,}{g}^0\right)$ . The pair $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\times \text{SO}\left(3\right)\mathrm{,}{S}^{4n+3}{\times }_{{\mathbb{Z}}_2}{S}^3\right)$ is a subgeometry of conformally flat pseudo-Riemannian homogeneous geometry $\left(\text{PO}\left(4n+\mathrm{4,4}\right)\mathrm{,}{S}^{4n+3}{\times }_{{\mathbb{Z}}_2}{S}^3\right)$ where $\text{PSp}\left(n+\mathrm{1,1}\right)\times \text{SO}\left(3\right)\le \text{PO}\left(4n+\mathrm{4,4}\right)$ .

Theorem B. A quaternionic 3 CR-manifold M is spherical (i.e. locally modeled on $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\mathrm{,}{S}^{4n+3}\right)$ ) if and only if the pseudo-Riemannian manifold $\left(M\times S^3\mathrm{,}g\right)$ is conformally flat, more precisely it is locally modeled on $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\times \text{SO}\left(3\right)\mathrm{,}{S}^{4n+3}{\times }_{{\mathbb{Z}}_2}{S}^3\right)$ .

We have constructed a conformal invariant on $\left(4n+3\right)$ -dimensional pseudo- conformal quaternionic CR manifolds in [3] . We think that the Weyl conformal curvature of our new pseudo-Riemannian metric obtained in Theorem A is theoretically the same as this invariant in view of Uniformization Theorem B. But we do not know whether they coincide.

Section 2 is a review of previous results and to give some definition of our notion. In Section 3 we prove the conformal equivalence of our pseudo-Riemannian metrics and prove Theorem A. In Section 4 first we relate our spherical 3 CR-homogeneous model $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\mathrm{,}{S}^{4n+3}\right)$ and the conformally flat pseudo-Riemannian homogeneous model $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\times \text{SO}\left(3\right)\mathrm{,}{S}^{4n+\mathrm{3,3}}\right)$ . We study properties of 3-dimensional lightlike groups with respect to the pseudo- Riemannian metric $g^0$ of type $\left(4n+\mathrm{3,3}\right)$ on $S^{4n+3}\times S^3$ . We apply these results to prove Theorem B.

2. Preliminaries

Let $\left(M,{\left\{{\omega }_{\alpha },J_{\alpha }\right\}}_{\alpha =1,2,3}\right)$ be a (4n + 3)-dimensional hypercomplex 3 CR-manifold. Put $\left({\omega }_{\alpha }\mathrm{,}{J}_{\alpha }\right)=\left(\omega \mathrm{,}J\right)$ for one of α’s. By the definition, $\left(M\mathrm{,}\left\{\omega \mathrm{,}J\right\}\right)$ is a CR-manifold. Let $C^{2n+\mathrm{2,0}}\left(M\right)$ be the canonical bundle over M (i.e. the $ℂ$ -line bundle of complex $\left(2n+\mathrm{2,0}\right)$ -forms). Put $C\left(M\right)=C^{2n+\mathrm{2,0}}\left(M\right)-\left\{0\right\}\mathrm{<mo>\; /\; </mo>}{ℝ}^{\mathrm{<mo>\; *\; </mo>}}$ which is a principal bundle: ${\text{S}}^1\to C\left(M\right)\stackrel{p}{\to }M$ . Compare [ [5] , Section 2.2]. Fefferman [6] has shown that $C\left(M\right)$ admits a Lorentz metric g for which the Lorentz isometries ${\text{S}}^1$ induce a lightlike vector field. We recognize the following definition from pseudo-Riemannian geometry.

Definition 1. In general if $S^1$ induces a lightlike vector field with respect to a Lorentz metric of a Lorentz manifold, then $S^1$ is said to be a lightlike group acting as Lorentz isometries. Similarly if each generator $S^1$ of $S^3$ is chosen to be a lightlike group, then we call $S^3$ also a lightlike group.

We recall a construction of the Fefferman-Lorentz metric from [5] (cf. [6] ). Let $\xi$ be the Reeb vector field for $\left(\omega \mathrm{,}J\right)$ . The circle ${\text{S}}^1$ generates the vector field $\text{T}$ on $C\left(M\right)$ . Define $dt$ to be a 1-form on $C\left(M\right)$ such that

$dt\left(\text{T}\right)=\mathrm{1,}dt\left(V\right)=0\left({}^{\forall }V\in TM\right)\mathrm{.}$ (2.1)

In [ [5] , (3.4) Proposition] J. Lee has shown that there exists a unique real 1-form $\sigma$ on $C\left(M\right)$ . The explicit form of $\sigma$ is obtained from [ [5] , (5.1) Theorem] in this case:

$\sigma =\frac{1}{2n+3}\left(dt+i{\omega }_{\alpha }^{\alpha }-\frac{i}{2}{h}^{\alpha \bar{\beta }}d{h}_{\alpha \bar{\beta }}-\frac{1}{2\left(2n+2\right)}R\omega \right).$ (2.2)

Here 1-forms $\left\{{\omega }_{\alpha }^{\beta }\mathrm{,}{\tau }_{\beta }\right\}$ are connection forms of $\omega$ such that

$\begin{array}{l}d\omega =i{h}_{\alpha \beta }{\omega }^{\alpha }\wedge {\omega }^{\bar{\beta }}\mathrm{,}\\ d{\omega }^{\alpha }={\omega }^{\beta }\wedge {\omega }_{\beta }^{\alpha }+\omega \wedge {\tau }^{\alpha }\mathrm{.}\end{array}$ (2.3)

The function R is the Webster scalar curvature on M. Note from (2.2)

$d\sigma =\frac{1}{2n+3}\left(id{\omega }_{\alpha }^{\alpha }-\frac{1}{2\left(2n+2\right)}Rd\omega -\frac{1}{2\left(2n+2\right)}dR\wedge \omega \right)\mathrm{.}$ (2.4)

Normalize $dt$ so that we may assume $\sigma \left(\text{T}\right)=1$ . Let $\sigma \odot \omega$ denote the symmetric 2-form defined by $\sigma \cdot \omega +\omega \cdot \sigma$ . Since $\omega \left(\text{T}\right)=0$ , it follows $\sigma \odot \omega \left(\text{T}\mathrm{,}\text{T}\right)=0$ . The Fefferman-Lorentz metric for $\left(\omega \mathrm{,}J\right)$ on $C\left(M\right)$ is defined by

$g\left(X\mathrm{,}Y\right)=\sigma \odot \omega \left(X\mathrm{,}Y\right)+d\omega \left(JX\mathrm{,}Y\right)\mathrm{.}$ (2.5)

Here $T\left(C\left(M\right)\right)=\langle \text{T}\rangle \oplus \langle \xi \rangle \oplus \text{ker}\omega$ . Since $\xi$ is the Reeb field, $d\omega \left(JX\mathrm{,}\xi \right)=0$ . As $\left[\text{ker}\omega \mathrm{,}\text{T}\right]=0$ , $d\omega \left(JX\mathrm{,}T\right)=0$ $\left({}^{\forall }X\in \text{k}er\omega \right)$ . On the other hand, $J\left(\left\{\text{T}\mathrm{,}\xi \right\}\right)=0$ by the definition. We have

$g\left(\xi \mathrm{,}\text{T}\right)=\mathrm{1,}g\left(\text{T}\mathrm{,}\text{T}\right)=0.$ (2.6)

Thus g becomes a Lorentz metric on $C\left(M\right)$ in which ${\text{S}}^1$ is a lightlike group.

Theorem 2 ( [5] ). If ${\omega }^{\prime }=u\omega$ , then $g^{\prime }=ug$ .

3. Hypercomplex 3 CR-Structure

Our strategy is as follows: first we construct a pseudo-Riemannian metric locally on each neighborhood of $M\times S^3$ by Condition I below and then sew these metrics on each intersection to get a globally defined pseudo-Riemannian metric on $M\times S^3$ using Theorem 4. (See the proof of Theorem A.)

Suppose that $\left(M,{\left\{{\omega }_{\alpha },J_{\alpha }\right\}}_{\alpha =1,2,3}\right)$ is a hypercomplex 3 CR-manifold of dimension $\left(4n+3\right)$ . Put $\omega ={\omega }_{1}i+{\omega }_{2}j+{\omega }_{3}k$ . It is an $\text{Im}ℍ$ -valued 1-form annihilating $\text{D}$ . In general, there is no canonical choice of $\omega$ annihilating $\text{D}$ . In [ [3] , Lemma 1.3] we observed that if ${\omega }^{\prime }$ is another $\text{Im}ℍ$ -valued 1-form annihilating $\text{D}$ , then

${\omega }^{\prime }=\lambda \omega \bar{\lambda }$ (3.1)

for some $ℍ$ -valued function $\lambda$ on M. (Here $\bar{\lambda }$ is the quaternion conjugate.) If we put $\lambda =\sqrt{u}a$ for a positive function u and $a\in \text{Sp}\left(1\right)$ , then ${\omega }^{\prime }=ua\omega \bar{a}$ such that the map $z\mapsto az\bar{a}$ $\left(z\in ℍ\right)$ represents a matrix function $A\in \text{SO}\left(3\right)$ . If ${\left\{{J^{\prime }}_{\alpha }\right\}}_{\alpha =1,2,3}$ is a hypercomplex structure on $\text{D}$ for ${\omega }^{\prime }$ , then they are related as $\left[{J^{\prime }}_1{J^{\prime }}_2{J^{\prime }}_3\right]=\left[J_1{J}_2{J}_3\right]A$ .

For each $\left({\omega }_{\alpha }\mathrm{,}{J}_{\alpha }\right)$ , we obtain a unique real 1-form ${\sigma }_{\alpha }$ on $C\left(M\right)$ from Section 2 (cf. (2.2)). First of all we construct a pseudo-Riemannian metric on $M\times S^3$ . In general $C\left(M\right)$ is a nontrivial principal ${\text{S}}^1$ -bundle. It is the trivial bundle when we restrict to a neighborhood. So for our use we assume:

Condition I. $C\left(M\right)$ is trivial as bundle, i.e. $C\left(M\right)=M\times S^1$ .

We construct a 1-form ${\sigma }_{\alpha }$ on $M\times S^3$ $\left(\alpha =1,2,3\right)$ as follows. Let ${\text{T}}_{\alpha }\mathrm{,}{\text{T}}_{\beta }\mathrm{,}{\text{T}}_{\gamma }$ generate ${\left\{e^{i\theta }\right\}}_{\theta \in ℝ}$ , ${\left\{e^{j\theta }\right\}}_{\theta \in ℝ}$ , ${\left\{e^{k\theta }\right\}}_{\theta \in ℝ}$ of $S^3$ respectively. Obtained as in (2.2), we have ${\sigma }_{\alpha }$ ’s on each $C\left(M\right)=M\times S^1$ such that

${\sigma }_{\alpha }\left({\text{T}}_{\alpha }\right)=\mathrm{1,}{\sigma }_{\beta }\left({\text{T}}_{\beta }\right)=\mathrm{1,}{\sigma }_{\gamma }\left({\text{T}}_{\gamma }\right)=1.$

We then extend ${\sigma }_{\alpha }$ to $M\times S^3$ by setting

${\sigma }_{\alpha }\left({\text{T}}_{\beta }\right)={\sigma }_{\alpha }\left({\text{T}}_{\gamma }\right)=0$ (3.2)

Since $\left[{\text{T}}_{\beta }\mathrm{,}{\text{T}}_{\gamma }\right]=2{\text{T}}_{\alpha }$ on $T{S}^3$ ,

$d{\sigma }_{\alpha }\left({\text{T}}_{\beta }\mathrm{,}{\text{T}}_{\gamma }\right)=-\frac{1}{2}{\sigma }_{\alpha }\left(\left[{\text{T}}_{\beta }\mathrm{,}{\text{T}}_{\gamma }\right]\right)=-1=-2{\sigma }_{\beta }\wedge {\sigma }_{\gamma }\left({\text{T}}_{\beta }\mathrm{,}{\text{T}}_{\gamma }\right)$ . Note that for any

$p\in M$ ,

$d{\sigma }_{\alpha }+2{\sigma }_{\beta }\wedge {\sigma }_{\gamma }=0\text{on}\left\{p\right\}\times S^3\left(\left(\alpha \mathrm{,}\beta \mathrm{,}\gamma \right)~\left(\mathrm{1,2,3}\right)\right)\mathrm{.}$ (3.3)

On the other hand, we recall the following from [ [3] , Lemma 4.1].

Proposition 3. The following hold:

$d{\omega }_1\left(J_{1}X\mathrm{,}Y\right)=d{\omega }_2\left(J_{2}X\mathrm{,}Y\right)=d{\omega }_3\left(J_{3}X\mathrm{,}Y\right)\left({}^{\forall }X\mathrm{,}Y\in \text{D}\right)\mathrm{.}$

In particular $g^{\text{D}}=d{\omega }_{\alpha }\circ J_{\alpha }$ is a positive definite invariant symmetric bilinear form on $\text{D}$ ;

$g^{\text{D}}\left(X\mathrm{,}Y\right)=g^{\text{D}}\left(J_{\alpha }X\mathrm{,}{J}_{\alpha }Y\right)\mathrm{.}$

Choose a frame field $\left\{X_1\mathrm{,}\cdots \mathrm{,}{X}_{4n}\right\}$ on $\text{D}$ such that $J_{\alpha }{X}_j=X_{\alpha n+j}$ $\left(j=1,\cdots ,n\right)$ with $d{\omega }_{\alpha }\left(J_{\alpha }{X}_j\mathrm{,}{X}_k\right)={\delta }_{jk}$ . Let ${\theta }^i$ be the dual frame to $X_i$ $\left(i=1,\cdots ,4n\right)$ such that

$d{\omega }_{\alpha }\left(J_{\alpha }X\mathrm{,}Y\right)={\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{\theta }^i\left(X\right)\cdot {\theta }^i\left(Y\right)\left({}^{\forall }X\mathrm{,}Y\in \text{D}\right)\mathrm{.}$ (3.4)

Let ${\xi }_{\alpha }$ be the Reeb field for ${\omega }_{\alpha }$ respectively. There is a decomposition $T\left(M\times S^3\right)=TM\oplus \left\{{\text{T}}_{\alpha }\mathrm{,}{\text{T}}_{\beta }\mathrm{,}{\text{T}}_{\gamma }\right\}=\left\{{\xi }_1\mathrm{,}{\xi }_2\mathrm{,}{\xi }_3\right\}\oplus \text{D}\oplus \left\{{\text{T}}_{\alpha }\mathrm{,}{\text{T}}_{\beta }\mathrm{,}{\text{T}}_{\gamma }\right\}$ .

As before let $\sigma \odot \omega ={\displaystyle {\sum }_{\alpha =1}^3}\left({\sigma }_{\alpha }\cdot {\omega }_{\alpha }+{\omega }_{\alpha }\cdot {\sigma }_{\alpha }\right)$ be a symmetric 2-form. Define a pseudo-Riemannian metric on $M\times S^3$ by

$\begin{array}{c}g\left(X\mathrm{,}Y\right)={\displaystyle \underset{\alpha =1}{\overset{3}{\sum }}}\left({\sigma }_{\alpha }\left(X\right)\cdot {\omega }_{\alpha }\left(Y\right)+{\omega }_{\alpha }\left(X\right)\cdot {\sigma }_{\alpha }\left(Y\right)\right)+d{\omega }_{\alpha }\left(J_{\alpha }X\mathrm{,}Y\right)\\ =\sigma \odot \omega \left(X\mathrm{,}Y\right)+{\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{\theta }^i\cdot {\theta }^i\left(X\mathrm{,}Y\right)\mathrm{.}\end{array}$ (3.5)

As in (2.6) it follows that $g\left({\xi }_{\alpha }\mathrm{,}{\text{T}}_{\alpha }\right)=1$ , $g\left({\text{T}}_{\alpha }\mathrm{,}{\text{T}}_{\alpha }\right)=0$ . If we note ${\sigma }_{\alpha }\left({\xi }_{\alpha }\right)\ne 0$ , letting ${\eta }_{\alpha }={\xi }_{\alpha }-{\sigma }_{\alpha }\left({\xi }_{\alpha }\right){\text{T}}_{\alpha }$ , it follows $g\left({\eta }_{\alpha }\mathrm{,}{\eta }_{\alpha }\right)=0$ . So

$\left[\begin{array}{cc}g\left({\eta }_{\alpha }\mathrm{,}{\eta }_{\alpha }\right)& g\left({\eta }_{\alpha }\mathrm{,}{\text{T}}_{\alpha }\right)\\ g\left({\text{T}}_{\alpha }\mathrm{,}{\eta }_{\alpha }\right)& g\left({\text{T}}_{\alpha }\mathrm{,}{\text{T}}_{\alpha }\right)\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$\left(\alpha =1,2,3\right)$ . As $g\mathrm{<mo>\; |\; </mo>}\text{D}=g^{\text{D}}$ is positive definite from Proposition 3, g is a pseudo-Riemannian metric of type $\left(4n+\mathrm{4,3}\right)$ on $M\times S^3$ .

Theorem 4. Let $g^{\prime }$ be the pseudo-Riemannian metric on $M\times S^3$ corre- sponding to another $\text{Im}ℍ$ -valued 1-form ${\omega }^{\prime }$ on M representing $\left(\text{D}\mathrm{,}Q\right)$ , i.e. ${\omega }^{\prime }=ua\omega \bar{a}$ $\left(a\in \text{Sp}\left(1\right)\mathrm{,}u>0\right)$ , then $g^{\prime }=u\cdot g$ .

We divide a proof according to whether ${\omega }^{\prime }=u\omega$ or ${\omega }^{\prime }=a\omega \bar{a}$ .

Proposition 5. If ${\omega }^{\prime }=u\omega$ , then $g^{\prime }=u\cdot g$ .

Proof. (Existence.) Suppose ${\omega }^{\prime }=u\omega$ . We show the existence of such a 1-form ${\sigma }^{\prime }$ for ${\omega }^{\prime }$ . Let ${\left\{{\text{T}}_{\alpha }\mathrm{,}{\xi }_{\alpha }\mathrm{,}{X}_1\mathrm{,}\cdots \mathrm{,}{X}_{4n}\right\}}_{\alpha =\mathrm{1,2,3}}$ be the frame on $M\times S^3$ for $\omega$ . Then ${\omega }^{\prime }$ determines another frame $\left\{{\text{<msup> <mo> T′ </mo> </msup>}}_{\alpha }\mathrm{,}{{\xi }^{\prime }}_{\alpha }\mathrm{,}{X^{\prime }}_1\mathrm{,}\cdots \mathrm{,}{X^{\prime }}_{4n}\right\}$ . Since each ${\text{<msup> <mo> T′ </mo> </msup>}}_{\alpha }$ generates the same $S^1$ as that of ${\text{T}}_{\alpha }$ , note

${\text{T}}_{\alpha }={\text{<msup> <mo> T′ </mo> </msup>}}_{\alpha }\left(\alpha =\mathrm{1,2,3}\right)\mathrm{.}$ (3.6)

Let ${\left\{X_i\right\}}_{i=1,\cdots ,4n}$ be the frame on $\text{D}$ . Then the Reeb field ${{\xi }^{\prime }}_{\alpha }$ for each ${{\omega }^{\prime }}_{\alpha }$ is described as

${\xi }_{\alpha }=u\cdot {{\xi }^{\prime }}_{\alpha }+x_1^{\left(\alpha \right)}\sqrt{u}{X^{\prime }}_1+\cdots +x_{4n}^{\left(\alpha \right)}\sqrt{u}{X^{\prime }}_{4n}\left(\alpha =\mathrm{1,2,3}\right)\mathrm{.}$ (3.7)

$\left({}^{\exists }{x}_i^{\left(\alpha \right)}\in ℝ\mathrm{,}i=\mathrm{1,}\cdots \mathrm{,}n\right)$ . As $u\cdot d\omega =d{\omega }^{\prime }$ on $\text{D}$ and $g^{\text{D}}\left(X\mathrm{,}Y\right)=g^{\text{D}}\left(J_{\alpha }X\mathrm{,}{J}_{\alpha }Y\right)$ from Proposition 3, there exists a matrix $B=\left(b_i^k\right)\in \text{Sp}\left(n\right)$ such that

$X_i=\sqrt{u}{\displaystyle \underset{k=1}{\overset{4n}{\sum }}}{b}_i^k{X^{\prime }}_k.$ (3.8)

Two frames $\left\{{\text{T}}_{\alpha }\mathrm{,}{\xi }_{\alpha }\mathrm{,}{X}_1\mathrm{,}\cdots \mathrm{,}{X}_{4n}\right\}$ , $\left\{{\text{<msup> <mo> T′ </mo> </msup>}}_{\alpha }\mathrm{,}{{\xi }^{\prime }}_{\alpha }\mathrm{,}{X^{\prime }}_1\mathrm{,}\cdots \mathrm{,}{X^{\prime }}_{4n}\right\}$ give the coframes $\left\{{\omega }_{\alpha }\mathrm{,}{\theta }^1\mathrm{,}\cdots \mathrm{,}{\theta }^{4n}\mathrm{,}{\sigma }_{\alpha }\right\}$ , $\left\{{{\omega }^{\prime }}_{\alpha }\mathrm{,}{{\theta }^{\prime }}^1\mathrm{,}\cdots \mathrm{,}{{\theta }^{\prime }}^{4n}\mathrm{,}{{\sigma }^{\prime }}_{\alpha }\right\}$ on $M\times S^3$ respectively. Then the above Equations (3.6), (3.7), (3.8) determine the relations between coframes:

$\begin{array}{l}{{\omega }^{\prime }}_{\alpha }=u\cdot {\omega }_{\alpha }\left(\alpha =\mathrm{1,2,3}\right)\mathrm{,}\\ {{\theta }^{\prime }}^i=\sqrt{u}{\displaystyle \underset{j=1}{\overset{4n}{\sum }}}{b}_j^i{\theta }^j+\sqrt{u}{x}_i^{\left(1\right)}\cdot {\omega }_1+\sqrt{u}{x}_i^{\left(2\right)}\cdot {\omega }_2+\sqrt{u}{x}_i^{\left(3\right)}\cdot {\omega }_3\mathrm{,}\end{array}$ (3.9)

Moreover if we put

$\begin{array}{l}{{\sigma }^{\prime }}_{\alpha }={\sigma }_{\alpha }-({\displaystyle \underset{j=1}{\overset{4n}{\sum }}}\left({\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{b}_j^i{x}_i^{\left(\alpha \right)}\right){\theta }^j+\frac{1}{2}{\displaystyle \underset{i\mathrm{<mo>\; =\; </mo>\; 1}}{\overset{4n}{\sum }}}{x}_i^{\left(\beta \right)}{x}_i^{\left(\alpha \right)}\cdot {\omega }_{\beta }\\ +\frac{1}{2}{\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{x}_i^{\left(\gamma \right)}{x}_i^{\left(\alpha \right)}\cdot {\omega }_{\gamma })-\frac{1}{2}{\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{\left|x_i^{\left(\alpha \right)}\right|}^2{\omega }_{\alpha }\mathrm{,}\end{array}$ (3.10)

then (3.15) and (3.10) show that

$\left({{\omega }^{\prime }}_1\mathrm{,}{{\omega }^{\prime }}_2\mathrm{,}{{\omega }^{\prime }}_3\mathrm{,}{{\theta }^{\prime }}^1\mathrm{,}\cdots \mathrm{,}{{\theta }^{\prime }}^{4n}\mathrm{,}{{\sigma }^{\prime }}_1\mathrm{,}{{\sigma }^{\prime }}_2\mathrm{,}{{\sigma }^{\prime }}_3\right)=\left({\omega }_1\mathrm{,}{\omega }_2\mathrm{,}{\omega }_3\mathrm{,}{\theta }^1\mathrm{,}\cdots \mathrm{,}{\theta }^{4n}\mathrm{,}{\sigma }_1\mathrm{,}{\sigma }_2\mathrm{,}{\sigma }_3\right)\text{P}$

for which

$\text{P}=\left(\begin{array}{ccccc}& \sqrt{u}{x}^{\left(1\right)}& \frac{-{\left|x^{\left(1\right)}\right|}^2}{2}& \frac{-x^{\left(1\right)}\cdot x^{\left(2\right)}}{2}& \frac{-x^{\left(1\right)}\cdot x^{\left(3\right)}}{2}\\ u{\text{I}}_3& \sqrt{u}{x}^{\left(2\right)}& \frac{-x^{\left(2\right)}\cdot x^{\left(1\right)}}{2}& \frac{-{\left|x^{\left(2\right)}\right|}^2}{2}& \frac{-x^{\left(2\right)}\cdot x^{\left(3\right)}}{2}\\ & \sqrt{u}{x}^{\left(3\right)}& \frac{-x^{\left(3\right)}\cdot x^{\left(1\right)}}{2}& \frac{-x^{\left(3\right)}\cdot x^{\left(2\right)}}{2}& \frac{-{\left|x^{\left(3\right)}\right|}^2}{2}\\ 0& \sqrt{u}B& -B{}^t{x}^{\left(1\right)}& -B{}^t{x}^{\left(2\right)}& -B{}^t{x}^{\left(3\right)}\\ 0& 0& & {\text{I}}_3& \end{array}\right)\mathrm{.}$

If ${\text{I}}_{4n}^3$ is a symmetric matrix defined by

${\text{I}}_{4n}^3=\left(\begin{array}{ccccc}0& 0& \cdots & 0& {\text{I}}_3\\ 0& & & & 0\\ ⋮& & {\text{I}}_{4n}& & ⋮\\ 0& & & & 0\\ {\text{I}}_3& 0& \cdots & 0& 0\end{array}\right)\mathrm{,}$ (3.11)

it is easily checked that $\text{P}{\text{I}}_{4n}^3{}^t\text{P}=u\cdot {\text{I}}_{4n}^3$ .

Letting ${\omega }^{\prime }=\left({{\omega }^{\prime }}_1\mathrm{,}{{\omega }^{\prime }}_2\mathrm{,}{{\omega }^{\prime }}_3\right)$ and ${\sigma }^{\prime }=\left({{\sigma }^{\prime }}_1,{{\sigma }^{\prime }}_2,{{\sigma }^{\prime }}_3\right)$ , we define a pseudo- Riemannian metric

$g^{\prime }={\sigma }^{\prime }\odot {\omega }^{\prime }+{\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{{\theta }^{\prime }}^i\cdot {{\theta }^{\prime }}^i\mathrm{.}$ (3.12)

Then a calculation shows

$\begin{array}{c}{g}^{\prime }={\displaystyle \underset{\alpha =1}{\overset{3}{\sum }}}\left({{\sigma }^{\prime }}_{\alpha }\cdot {{\omega }^{\prime }}_{\alpha }+{{\omega }^{\prime }}_{\alpha }\cdot {{\sigma }^{\prime }}_{\alpha }\right)+{\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{{\theta }^{\prime }}^i\cdot {{\theta }^{\prime }}^i\\ =\left({\omega }^{\prime }\mathrm{,}{{\theta }^{\prime }}^1\mathrm{,}\cdots \mathrm{,}{{\theta }^{\prime }}^{4n}\mathrm{,}{\sigma }^{\prime }\right){\text{I}}_{4n}^3{}^t\left({\omega }^{\prime }\mathrm{,}{{\theta }^{\prime }}^1\mathrm{,}\cdots \mathrm{,}{{\theta }^{\prime }}^{4n}\mathrm{,}{\sigma }^{\prime }\right)\\ =\left(\omega \mathrm{,}{\theta }^1\mathrm{,}\cdots \mathrm{,}{\theta }^{2n}\mathrm{,}\sigma \right)\text{P}{\text{I}}_{4n}^3{}^t\text{P}{}^t\left(\omega \mathrm{,}{\theta }^1\mathrm{,}\cdots \mathrm{,}{\theta }^{2n}\mathrm{,}\sigma \right)\\ =u\cdot \left(\omega \mathrm{,}{\theta }^1\mathrm{,}\cdots \mathrm{,}{\theta }^{2n}\mathrm{,}\sigma \right){\text{I}}_{4n}^3{}^t\left(\omega \mathrm{,}{\theta }^1\mathrm{,}\cdots \mathrm{,}{\theta }^{2n}\mathrm{,}\sigma \right)\\ =u\left({\displaystyle \underset{\alpha =1}{\overset{3}{\sum }}}\left({\sigma }_{\alpha }\cdot {\omega }_{\alpha }+{\omega }_{\alpha }\cdot {\sigma }_{\alpha }\right)+{\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{\theta }^i\cdot {\theta }^i\right)=u\cdot g\mathrm{.}\end{array}$ (3.13)

(Uniqueness.) We prove the above ${\sigma }^{\prime }$ is uniquely determined with respect to ${\omega }^{\prime }$ . Let $\mathcal{F}=\left\{{\omega }_{\alpha }\mathrm{,}{\theta }^1\mathrm{,}\cdots \mathrm{,}{\theta }^{4n}\mathrm{,}{\theta }^{4n+1}\mathrm{,}{\theta }^{4n+2}\right\}$ be the coframe for ${\omega }_{\alpha }$ where ${\theta }^{4n+1}={\omega }_{\beta },{\theta }^{4n+2}={\omega }_{\gamma }$ . We have a Fefferman-Lorentz metric on $M\times S^1$ from (3.5) and (3.4) under Condition I:

$\begin{array}{c}{g}_{\alpha }={\sigma }_{\alpha }\odot {\omega }_{\alpha }+\frac{1}{3}d{\omega }_{\alpha }\circ J_{\alpha }\\ ={\sigma }_{\alpha }\odot {\omega }_{\alpha }+\frac{1}{3}\left({\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{\theta }^i\cdot {\theta }^i+{\omega }_{\beta }\cdot {\omega }_{\beta }+{\omega }_{\gamma }\cdot {\omega }_{\gamma }\right)\mathrm{.}\end{array}$ (3.14)

(We take the coefficient $\frac{1}{3}$ for our use.) When ${{\omega }^{\prime }}_{\alpha }=u{\omega }_{\alpha }$ , the coframe $\mathcal{F}$

will be transformed into a coframe ${\mathcal{F}}^{\prime }=\left\{{{\omega }^{\prime }}_{\alpha }\mathrm{,}{{\theta }^{\prime }}_{\alpha }^1\mathrm{,}\cdots \mathrm{,}{{\theta }^{\prime }}_{\alpha }^{4n}\mathrm{,}{{\theta }^{\prime }}_{\alpha }^{4n+1}\mathrm{,}{{\theta }^{\prime }}_{\alpha }^{4n+2}\right\}$ such as

$\begin{array}{l}{{\theta }^{\prime }}_{\alpha }^i=\sqrt{u}{\displaystyle \underset{j}{\sum }}{c}_{\alpha j}^i{\theta }^j+\sqrt{u}{y}_{\alpha }^i{\omega }_{\alpha },\\ {{\theta }^{\prime }}_{\alpha }^{4n+1}=\sqrt{u}{\theta }^{4n+1}=\sqrt{u}{\omega }_{\beta },\\ {{\theta }^{\prime }}_{\alpha }^{4n+2}=\sqrt{u}{\theta }^{4n+2}=\sqrt{u}{\omega }_{\gamma },\end{array}$ (3.15)

$\left({}^{\exists }{y}_{\alpha }^i\in ℝ\mathrm{,}{}^{\exists }\left(c_{\alpha j}^i\right)\in \text{Sp}\left(n\right)\mathrm{,}i\mathrm{,}j=\mathrm{1,}\cdots \mathrm{,}n\right)$ .

If ${g^{\prime }}_{\alpha }$ is the corresponding metric on $M\times S^1$ , then ${g^{\prime }}_{\alpha }=u{g}_{\alpha }$ by Theorem 2 and there exists a unique 1-form ${\tilde{\sigma }}_{\alpha }$ such that

$\begin{array}{c}{g^{\prime }}_{\alpha }={\tilde{\sigma }}_{\alpha }\odot {{\omega }^{\prime }}_{\alpha }+\frac{1}{3}\left({\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{{\theta }^{\prime }}_{\alpha }^i\cdot {{\theta }^{\prime }}_{\alpha }^i+{{\theta }^{\prime }}_{\alpha }^{4n+1}\cdot {{\theta }^{\prime }}_{\alpha }^{4n+1}+{{\theta }^{\prime }}_{\alpha }^{4n+2}\cdot {{\theta }^{\prime }}_{\alpha }^{4n+2}\right)\\ ={\tilde{\sigma }}_{\alpha }\odot {{\omega }^{\prime }}_{\alpha }+\frac{1}{3}\left({\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{{\theta }^{\prime }}_{\alpha }^i\cdot {{\theta }^{\prime }}_{\alpha }^i+u{\omega }_{\beta }\cdot {\omega }_{\beta }+u{\omega }_{\gamma }\cdot {\omega }_{\gamma }\right)\mathrm{.}\end{array}$ (3.16)

If we sum up this equality for $\alpha =1,2,3$ ;

$\begin{array}{c}{g^{\prime }}_1+{g^{\prime }}_2+{g^{\prime }}_3=\tilde{\sigma }\odot {\omega }^{\prime }+\frac{1}{3}{\displaystyle \underset{\alpha \mathrm{,}i}{\sum }}{{\theta }^{\prime }}_{\alpha }^i\cdot {{\theta }^{\prime }}_{\alpha }^i+\frac{2}{3}u\left({\omega }_{\alpha }\cdot {\omega }_{\alpha }+{\omega }_{\beta }\cdot {\omega }_{\beta }+{\omega }_{\gamma }\cdot {\omega }_{\gamma }\right)\\ =u{g}_1+u{g}_2+u{g}_3\\ =u\left(\sigma \odot \omega +{\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{\theta }^i\cdot {\theta }^i+\frac{2}{3}\left({\omega }_{\alpha }\cdot {\omega }_{\alpha }+{\omega }_{\beta }\cdot {\omega }_{\beta }+{\omega }_{\gamma }\cdot {\omega }_{\gamma }\right)\right)\mathrm{,}\end{array}$

which yields

$\tilde{\sigma }\odot {\omega }^{\prime }+\frac{1}{3}{\displaystyle \underset{\alpha =1}{\overset{3}{\sum }}}{\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{{\theta }^{\prime }}_{\alpha }^i\cdot {{\theta }^{\prime }}_{\alpha }^i=u\left(\sigma \odot \omega +{\displaystyle \underset{i=1}{\overset{4n}{\sum }}}{\theta }^i\cdot {\theta }^i\right)=ug\mathrm{.}$ (3.17)

Compared this with (3.13) it follows

${\sigma }^{\prime }=\tilde{\sigma }\mathrm{,}i\mathrm{.}e\mathrm{.}{{\sigma }^{\prime }}_{\alpha }={\tilde{\sigma }}_{\alpha }\left(\alpha =\mathrm{1,2,3}\right)\mathrm{.}$ (3.18)

By uniqueness of ${\tilde{\sigma }}_{\alpha }$ , ${{\sigma }^{\prime }}_{\alpha }$ defined by (3.10) is a unique real 1-form with respect to ${\omega }^{\prime }$ .

Next put $\tilde{\omega }=a\cdot \omega \cdot \bar{a}$ . The conjugate $z\mapsto az\bar{a}$ $\left({}^{\forall }z\in ℍ\right)$ represents a

matrix $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]\in \text{SO}\left(3\right)$ . Then it follows

$\tilde{\omega }=\left[{\omega }_1\mathrm{,}{\omega }_2\mathrm{,}{\omega }_3\right]A\left[\begin{array}{c}i\\ j\\ k\end{array}\right]$ (3.19)

By our definition, a hypercomplex structure $\left\{J_1\mathrm{,}{J}_2\mathrm{,}{J}_3\right\}$ on $\text{D}$ satisfies that ${\left(d{\omega }_{\beta }\mathrm{<mo>\; |\; </mo>}\text{D}\right)}^{-1}\circ \left(d{\omega }_{\alpha }\mathrm{<mo>\; |\; </mo>}\text{D}\right)=J_{\gamma }$ $\left(\alpha \mathrm{,}\beta \mathrm{,}\gamma \right)~\left(\mathrm{1,2,3}\right)$ . A new hypercomplex structure on $\text{D}$ is described as

$\left(\begin{array}{c}{\tilde{J}}_1\\ {\tilde{J}}_2\\ {\tilde{J}}_3\end{array}\right)={}^{t}A\left(\begin{array}{c}{J}_1\\ J_2\\ J_3\end{array}\right)\mathrm{.}$ (3.20)

Differentiate (3.19) and restrict to $\text{D}$ (in fact, $d\tilde{\omega }=a\cdot d\omega \cdot \bar{a}$ on $\text{D}$ ), using Proposition 3, a calculation shows

$\begin{array}{c}d{\tilde{\omega }}_{\alpha }\left(X\mathrm{,}Y\right)=-a_{1\alpha }{g}^{\text{D}}\left(J_{1}X\mathrm{,}Y\right)+a_{2\alpha }{g}^{\text{D}}\left(J_{2}X\mathrm{,}Y\right)+a_{3\alpha }{g}^{\text{D}}\left(J_{3}X\mathrm{,}Y\right)\\ =-g^{\text{D}}\left(\left(a_{1\alpha }{J}_1+a_{2\alpha }{J}_2+a_{3\alpha }{J}_3\right)X\mathrm{,}Y\right)=-g^{\text{D}}\left({\tilde{J}}_{\alpha }X\mathrm{,}Y\right)\mathrm{,}\end{array}$

$d{\tilde{\omega }}_{\alpha }\left({\tilde{J}}_{\alpha }X\mathrm{,}Y\right)=g^{\text{D}}\left(X\mathrm{,}Y\right)\left(\alpha =\mathrm{1,2,3}\right)\mathrm{.}$ (3.21)

In particular, we have ${\left(d{\tilde{\omega }}_{\beta }\mathrm{<mo>\; |\; </mo>}\text{D}\right)}^{-1}\circ \left(d{\tilde{\omega }}_{\alpha }\mathrm{<mo>\; |\; </mo>}\text{D}\right)={\tilde{J}}_{\gamma }$ $\left(\alpha \mathrm{,}\beta \mathrm{,}\gamma \right)~\left(\mathrm{1,2,3}\right)$ .

Proposition 6. If $\tilde{\omega }=a\omega \bar{a}$ , then $\tilde{g}=g$ .

Proof. Let $\tilde{g}\left(X\mathrm{,}Y\right)=\tilde{\sigma }\odot \tilde{\omega }\left(X\mathrm{,}Y\right)+d{\tilde{\omega }}_{\alpha }\left({\tilde{J}}_{\alpha }X\mathrm{,}Y\right)$ . Since ${\tilde{\sigma }}_{\alpha }$ is uniquely determined by ${\tilde{\omega }}_{\alpha }$ and $\tilde{\omega }=\left[{\omega }_1\mathrm{,}{\omega }_2\mathrm{,}{\omega }_3\right]A=\omega A$ from (3.19), it implies that

$\tilde{\sigma }=\left[{\sigma }_1,{\sigma }_2,{\sigma }_3\right]A=\sigma A.$ (3.22)

Note that

$\begin{array}{c}\tilde{\sigma }\odot \tilde{\omega }={\displaystyle \underset{\alpha =1}{\overset{3}{\sum }}}\left({\tilde{\sigma }}_{\alpha }\cdot {\tilde{\omega }}_{\alpha }+{\tilde{\omega }}_{\alpha }\cdot {\tilde{\sigma }}_{\alpha }\right)=\sigma A{}^{t}A{}^t\omega +\omega A{}^{t}A{}^t\sigma \\ =\sigma {}^t\omega +\omega {}^t\sigma =\sigma \odot \omega \mathrm{.}\end{array}$ (3.23)

By (3.21),

$\tilde{g}=\tilde{\sigma }\odot \tilde{\omega }+d{\tilde{\omega }}_{\alpha }\circ {\tilde{J}}_{\alpha }=\sigma \odot \omega +g^{\text{D}}=g\mathrm{.}$

Proof of Theorem 4. Suppose ${\omega }^{\prime }=\lambda \omega \bar{\lambda }=u\tilde{\omega }$ where $\tilde{\omega }=a\omega \bar{a}$ . It follows from Proposition 5 that $g^{\prime }=u\tilde{g}$ . By Proposition 6, we have $\tilde{g}=g$ and hence $g^{\prime }=ug$ . This finishes the proof under Condition I.

Proof of Theorem A

Proof. Let $\left(M\mathrm{,}\left\{\text{D}\mathrm{,}Q\right\}\right)$ be a quaternionic 3 CR-manifold. Then M has an open cover ${\left\{U_i\right\}}_{i\in \Lambda }$ where each $U_i$ admits a hypercomplex 3 CR-structure ${\left({\omega }_{\alpha }^{\left(i\right)}\mathrm{,}{J}_{\alpha }^{\left(i\right)}\right)}_{\alpha =\mathrm{1,2,3}}$ . Put ${\omega }^{\left(i\right)}={\omega }_1^{\left(i\right)}i+{\omega }_2^{\left(i\right)}j+{\omega }_3^{\left(i\right)}k$ which is an $\text{Im}ℍ$ -valued 1-form on $U_i$ . Since we may assume that $U_i$ is homeomorphic to a ball (i.e. contractible), Condition I is satisfied for each $U_i$ , i.e. $C\left(U_i\right)=U_i\times S^1$ . Then we have a pseudo-Riemannian metric $g^{\left(i\right)}={{\displaystyle {\displaystyle \sum }}}_{\alpha =1}^3{\sigma }_{\alpha }^{\left(i\right)}\odot {\omega }_{\alpha }^{\left(i\right)}+d{\omega }_{\alpha }^{\left(i\right)}\circ J_{\alpha }^{\left(i\right)}$ on $U_i\times S^3$ for ${\omega }^{\left(i\right)}$ by Theorem 4. Suppose $U_i\cap U_j\ne \varnothing$ . By condition a) of 2) (cf. Introduction), $\text{D}\mathrm{<mo>\; |\; </mo>}{U}_i\cap U_j=\text{ker}{\omega }^{\left(i\right)}\mathrm{<mo>\; |\; </mo>}{U}_i\cap U_j=\text{ker}{\omega }^{\left(j\right)}\mathrm{<mo>\; |\; </mo>}{U}_i\cap U_j$ . Then by the equivalence (3.1) there exists a function $\lambda =\sqrt{u}a$ defined on $U_i\cap U_j$ such that

${\omega }^{\left(j\right)}=\lambda \cdot {\omega }^{\left(i\right)}\cdot \bar{\lambda }=ua{\omega }^{\mathrm{<mo\; stretchy="false">\; (\; </mo>}i\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\bar{a}\text{on}{U}_i\cap U_j\mathrm{.}$ (3.24)

It follows from Theorem 4 that $g^{\left(j\right)}=u{g}^{\left(i\right)}$ on $U_i\cap U_j$ . We may put $u=u^{ji}$ which is a positive function defined on $U_i\cap U_j$ . By construction, it is easy to see that $u^{ki}=u^{kj}{u}^{ji}$ on $U_i\cap U_j\cap U_k\ne \varnothing$ . This implies that ${\left\{u\right\}}_{i\mathrm{,}j\in \Lambda }$ defines a 1-cocycle on M. Since ${ℝ}^{+}$ is a fine sheaf as the germ of local continuous functions, note that the first cohomology $H^1\left(\mathcal{U}\mathrm{,}{ℝ}^{+}\right)=0$ . (Here $\mathcal{U}$ is a chain complex of covers running over all open covers of M.) Therefore there exists a local function ${\left\{f\right\}}_{i\mathrm{,}j\in \Lambda }$ defined on each $U_i$ such that $\delta f\left(j,i\right)=u^{ji}$ , i.e. $f_i\cdot f_j^{-1}=u^{ji}$ on $U_i\cap U_j$ . We obtain that

$f_j\cdot g^{\left(j\right)}=f_i\cdot g^{\left(i\right)}\text{on}\left(U_i\cap U_j\right)\times S^3\mathrm{.}$

Then we may define

$g\mathrm{<mo>\; |\; </mo>}{U}_i\times S^3=f_i\cdot g^{\left(i\right)}\mathrm{.}$ (3.25)

so that g is a globally defined pseudo-Riemannian metric on $M\times S^3$ . If another family ${\left\{{{\omega }^{\prime }}^i\right\}}_{i\in \Lambda }$ represents the same quaternionic 3 CR-structure $\left(\text{D}\mathrm{,}Q\right)$ , then the same argument shows that $g^{\prime }=ug$ on $M\times S^3$ for some positive function. Hence the conformal class $\left[g\right]$ is an invariant for quaternionic 3 CR-structure. In particular, the Weyl curvature tensor $W\left(g\right)$ is also an invariant. This completes the proof of Theorem A.

4. Model Geometry and Transformations

We introduce spherical 3 CR-homogeneous model $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\mathrm{,}{S}^{4n+3}\right)$ and conformally flat pseudo-Riemannian homogeneous model $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\times \text{SO}\left(3\right)\mathrm{,}{S}^{4n+\mathrm{3,3}}\right)$ equipped with pseudo-Riemannian metric $g^0$ of type $\left(4n+\mathrm{3,3}\right)$ and then characterize the lightlike subgroup in $\text{PSp}\left(n+\mathrm{1,1}\right)\times \text{SO}\left(3\right)$ .

4.1. Pseudo-Riemannian Metric g⁰

Let us start with the quaternionic vector space ${ℍ}^{n+2}$ endowed with the Her- mitian form:

$\langle z\mathrm{,}w\rangle ={\bar{z}}_1{w}_1+\cdots +z_{n+1}{w}_{n+1}-{\bar{z}}_{n+2}{w}_{n+2}\left(z\mathrm{,}w\in {ℍ}^{n+2}\right)\mathrm{.}$ (4.1)

The q-cone is defined by

$V_0=\left\{z\in {ℍ}^{n+2}-\left\{0\right\}\mathrm{<mo>\; |\; </mo>}\langle z\mathrm{,}z\rangle \rangle \mathrm{<mo>\; =\; </mo>\; 0}\right\}\mathrm{.}$ (4.2)

When ${ℍ}^{n+2}$ is viewed as the real vector space ${ℝ}^{4n+8}$ , $\text{O}\left(4n+\mathrm{4,4}\right)$ denotes the full subgroup of $\text{GL}\left(4n+\mathrm{8,}ℝ\right)$ preserving the bilinear form $\mathrm{Re}\langle ,\rangle$ . Consider the commutative diagrams below. The image of the pair $\left(\text{O}\left(4n+\mathrm{4,4}\right)\mathrm{,}{V}_0\right)$ by the projection $P_{�}$ is the homogeneous model of conformally flat pseudo-Riemannian geometry $\left(\text{PO}\left(4n+\mathrm{4,4}\right)\mathrm{,}{S}^{4n+\mathrm{3,3}}\right)$ in which $S^{4n+\mathrm{3,3}}=P_{ℝ}\left(V_0\right)$ is diffeomorphic to a quotient manifold $S^{4n+3}{\times }_{{\mathbb{Z}}_2}{S}^3$ . The identification ${ℍ}^{n+2}={ℝ}^{4n+8}$ gives a natural embedding $\text{Sp}\left(n+\mathrm{1,1}\right)\cdot \text{Sp}\left(1\right)\to \text{O}\left(4n+\mathrm{4,4}\right)$ which results a special geometry $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\times \text{SO}\left(3\right)\mathrm{,}{S}^{4n+\mathrm{3,3}}\right)$ from $\left(\text{PO}\left(4n+\mathrm{4,4}\right)\mathrm{,}{S}^{4n+\mathrm{3,3}}\right)$ .

As usual, the image of $\left(\text{Sp}\left(n+\mathrm{1,1}\right)\cdot \text{Sp}\left(1\right)\mathrm{,}{V}_0\right)$ by $P_{ℍ}$ is spherical quarter- nionic 3 CR-geometry $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\mathrm{,}{S}^{4n+3}\right)$ .

(4.3)

We describe a pseudo-Riemannian metric $g^0$ on $S^{4n+\mathrm{3,3}}=S^{4n+3}{\times }_{{\mathbb{Z}}_2}{S}^3$ . Let $S^{4n+3}\times S^3$ be the product of unit spheres. For $\left(z\mathrm{,}w\right)\in S^{4n+3}\times S^3$ , ${\left|z\right|}^2-{\left|w\right|}^2=1-1=0$ so $S^{4n+3}\times S^3\subset V_0$ . Then $P_{ℝ}\left(V_0\right)=S^{4n+\mathrm{3,3}}$ induces a 2-fold covering $P_{ℝ}\mathrm{<mo>\; :\; </mo>}{S}^{4n+3}\times S^3\to S^{4n+\mathrm{3,3}}$ for which $P_{\mathrm{<mo>\; *\; </mo>}}\mathrm{<mo>\; :\; </mo>}T\left(S^{4n+3}\times S^3\right)\to T{S}^{4n+\mathrm{3,3}}$ is an isomorphism.

Let $x\in S^{4n+3}\times S^3$ where we put $P_{ℝ}\left(x\right)=\left[x\right]$ . Choose $y\in S^{4n+3}\times S^3$ such that $\langle x,y\rangle =1$ . Denote by ${\left\{x\mathrm{,}y\right\}}^{\perp }$ the orthogonal complement in ${ℍ}^{n+2}$ with respect to $\langle ,\rangle$ . As $T_x{V}_0=\left\{Z\in {ℍ}^{n+2}|\mathrm{Re}\langle x\mathrm{,}Z\rangle =0\right\}$ , it follows $T_x{V}_0=y\mathrm{Im}ℍ\oplus xℍ\oplus {\left\{x\mathrm{,}y\right\}}^{\perp }\subset {ℍ}^{n+2}$ such that

$T_x\left(S^{4n+3}\times S^3\right)=y\mathrm{Im}ℍ\oplus x\mathrm{Im}ℍ\oplus {\left\{x\mathrm{,}y\right\}}^{\perp }\mathrm{.}$

In particular, $T_x{V}_0=xℝ\oplus T_x\left(S^{4n+3}\times S^3\right)$ . Note that this decomposition does not depend on the choice of points $x^{\prime }\in \left[x\right]$ and $y^{\prime }$ with $\langle x^{\prime },y^{\prime }\rangle =1$ . (see [3] , Theorem 6.1]). We define a pseudo-Riemannian metric on $S^{4n+\mathrm{3,3}}$ to be

$g_{\left[x\right]}^0\left(P_{ℝ*}X\mathrm{,}{P}_{ℝ*}Y\right)=\mathrm{Re}\langle X\mathrm{,}Y\rangle \left({}^{\forall }X\mathrm{,}Y\in T_x\left(S^{4n+3}\times S^3\right)\right)\mathrm{.}$ (4.4)

Noting $\mathrm{Re}\langle ya,ya\rangle =\mathrm{Re}\langle xa,xa\rangle =0$ , $\mathrm{Re}\langle xa,ya\rangle =1$ $\left({}^{\forall }a\in \text{Sp}\left(1\right)\right)$ and $\mathrm{Re}\langle ,\rangle {\mathrm{<mo>\; |\; </mo>}}_{{\left\{x\mathrm{,}y\right\}}^{\perp }}$ is positive definite, $g_{\left[x\right]}^0$ is a pseudo-Riemannian metric of type $\left(4n+\mathrm{3,3}\right)$ at each $\left[x\right]\in S^{4n+\mathrm{3,3}}$ .

4.2. Conformal Group $O\left(4n+4,4\right)$

It is known more or less but we need to check that $\text{O}\left(4n+4,4\right)$ acts on $S^{4n+3}\times S^3$ as conformal transformations with respect to $\mathrm{Re}\langle ,\rangle$ and so does $\text{PO}\left(4n+\mathrm{4,4}\right)$ on $\left(S^{4n+\mathrm{3,3}}\mathrm{,}{g}^0\right)$ .

For any $h\in \text{O}\left(4n+\mathrm{4,4}\right)$ , $\langle hx\mathrm{,}hx\rangle =\langle x\mathrm{,}x\rangle =0$ so $hx\in V_0$ . However $hx$ does not necessarily belong to $S^{4n+3}\times S^3$ . Normalized $hx$ , there is $x^{\prime }\in S^{4n+3}\times S^3$ such that $\left(hx\right)\lambda =x^{\prime }$ for some $\lambda \in {ℝ}^{+}$ . Note $\left[hx\right]=P_{ℝ}\left(hx\right)=P_{ℝ}\left(x^{\prime }\right)$ . If $R_{\lambda }\mathrm{<mo>\; :\; </mo>}{ℍ}^{n+2}\to {ℍ}^{n+2}$ is the right multiplication defined by $R_{\lambda }\left(z\right)=z\lambda$ , then there is the commutative diagram:

in which $R_{\mathrm{<mo>\; *\; </mo>}}\left(h_{\mathrm{<mo>\; *\; </mo>}}X\right)=\left(h_{\mathrm{<mo>\; *\; </mo>}}X\right)\lambda \in T_{x^{\prime }}{V}_0$ . As $T_{x^{\prime }}{V}_0=x^{\prime }ℝ\oplus T_{x^{\prime }}\left(S^{4n+3}\times S^3\right)$ , we have $\left(h_{\mathrm{<mo>\; *\; </mo>}}X\right)\lambda =x^{\prime }\mu +X^{\prime }$ for some $\mu \in ℝ$ , $X^{\prime }\in T_{x^{\prime }}\left(S^{4n+3}\times S^3\right)$ . Since $P_{\mathrm{<mo>\; *\; </mo>}}\left(T_x{ℝ}^{\mathrm{<mo>\; *\; </mo>}}\right)=P_{\mathrm{<mo>\; *\; </mo>}}\left(xℝ\right)=0$ and $P_{ℝ}\mathrm{<mo>\; :\; </mo>}\left(\text{O}\left(4n+\mathrm{4,4}\right)\mathrm{,}{V}_0\right)\to \left(\text{PO}\left(4n+\mathrm{4,4}\right)\mathrm{,}{S}^{4n+\mathrm{3,3}}\right)$ is equivariant, it follows

$h_{\mathrm{<mo>\; *\; </mo>}}{P}_{ℝ*}\left(X\right)=P_{ℝ*}\left(h_{\mathrm{<mo>\; *\; </mo>}}X\right)=P_{ℝ*}\left(\left(h_{\mathrm{<mo>\; *\; </mo>}}X\right)\lambda \right)=P_{ℝ*}\left(x^{\prime }\mu +X^{\prime }\right)=P_{ℝ*}\left(X^{\prime }\right)\mathrm{.}$

Similarly $h_{\mathrm{<mo>\; *\; </mo>}}{P}_{ℝ*}\left(Y\right)=P_{ℝ*}\left(Y^{\prime }\right)$ for $\left(h_{\mathrm{<mo>\; *\; </mo>}}Y\right)\lambda =x^{\prime }\nu +Y^{\prime }$ for some $\nu \in ℝ$ , $Y^{\prime }\in T_{x^{\prime }}\left(S^{4n+3}\times S^3\right)$ . As $\mathrm{Re}\langle x^{\prime },X^{\prime }\rangle =\mathrm{Re}\langle x^{\prime },Y^{\prime }\rangle =0$ , a calculation shows

$\begin{array}{l}{g}_{\left[hx\right]}^0\left(h_{*}{P}_{ℝ*}\left(X\right),h_{*}{P}_{ℝ*}\left(Y\right)\right)\\ =g_{\left[hx\right]}^0\left(P_{ℝ*}\left(X^{\prime }\right),P_{ℝ*}\left(Y^{\prime }\right)\right)=\mathrm{Re}\langle X^{\prime },Y^{\prime }\rangle \\ =\mathrm{Re}\langle x^{\prime }\mu +X^{\prime },x^{\prime }\nu +Y^{\prime }\rangle =\mathrm{Re}\langle \left(h_{*}X\right)\lambda ,\left(h_{*}Y\right)\lambda \rangle \\ ={\lambda }^2\mathrm{Re}\langle h_{*}X,h_{*}Y\rangle ={\lambda }^2\mathrm{Re}\langle X,Y\rangle \\ ={\lambda }^2{g}_{\left[x\right]}^0\left(P_{ℝ*}\left(X\right),P_{ℝ*}\left(Y\right)\right).\end{array}$

Hence $h\in \text{O}\left(4n+\mathrm{4,4}\right)$ acts as conformal transformation with respect to $g^0$ .

4.3. Conformal Subgroup $Sp\left(n+1,1\right)\cdot Sp(\; 1\; )$

Let $\left(I\mathrm{,}J\mathrm{,}K\right)$ be the standard hypercomplex structure on ${ℍ}^{n+2}$ defined by

$Iz=-zi,Jz=-zj,Kz=-zk.$

Put $Q=\text{span}\left(I\mathrm{,}J\mathrm{,}K\right)$ as the associated quaternionic structure. Then $\mathrm{Re}\langle ,\rangle$ leaves invariant Q. The full subgroup of $\text{O}\left(4n+\mathrm{4,4}\right)$ preserving Q is isomorphic to $\text{Sp}\left(n+\mathrm{1,1}\right)\cdot \text{Sp}\left(1\right)$ , i.e. the intersection of $\text{O}\left(4n+\mathrm{4,4}\right)$ with $\text{GL}\left(n+\mathrm{2,}ℍ\right)\cdot \text{GL}\left(\mathrm{1,}ℍ\right)$ .

Let $\rho \mathrm{<mo>\; :\; </mo>}{S}^3\to \text{O}\left(4n+\mathrm{4,4}\right)$ be a faithful representation. Then the subgroup $\rho \left(S^3\right)$ preserves Q so it is contained in

$\left(\stackrel{n+2}{\stackrel{︷}{\text{Sp}\left(1\right)\times \cdots \times \text{Sp}\left(1\right)}}\right)\cdot \text{Sp}\left(1\right)\le \stackrel{n+2}{\stackrel{︷}{\text{SO}\left(4\right)\times \cdots \times \text{SO}\left(4\right)}}$

which is a subgroup of $\text{SO}\left(4n+4\right)\times \text{SO}\left(4\right)$ .

4.4. Three Dimensional Lightlike Group

Choose $S^1\le S^3$ and consider a representation restricted to $S^1$ . As we may assume that the semisimple group $\rho \left(S^3\right)$ belongs to $\left(\text{Sp}\left(1\right)\times \cdots \times \text{Sp}\left(1\right)\right)\cdot \text{Sp}\left(1\right)$ , this reduces to a faithful representation: $\rho \mathrm{<mo>\; :\; </mo>}{S}^1\to T^{n+2}\cdot S^1$ such that

$\rho \left(t\right)=\left(\left({\text{e}}^{i{a}_{1}t},\cdots ,{\text{e}}^{i{a}_{n+2}t}\right)\cdot {\text{e}}^{ibt}\right).$ (4.5)

Here we may assume that $a_i\ge 0$ are relatively prime $\left(i=1,\cdots ,n+2\right)$ without loss of generality, and either $b=0$ or 1. The element $\rho \left(t\right)$ acts on $S^{4n+3}\times S^3\subset V_0$ as

$\begin{array}{c}\rho \left(t\right)\left(z_1\mathrm{,}\cdots \mathrm{,}{z}_{n+1}\mathrm{,}w\right)=\left(e^{i{a}_{1}t}{z}_1\mathrm{,}\cdots \mathrm{,}{e}^{i{a}_{n+1}t}{z}_{n+1}\mathrm{,}{e}^{i{a}_{n+2}t}w\right)\cdot e^{-ibt}\\ =\left(e^{i{a}_{1}t}{z}_1{e}^{-ibt}\mathrm{,}\cdots \mathrm{,}{e}^{i{a}_{n+1}t}{z}_{n+1}{e}^{-ibt}\mathrm{,}{e}^{i{a}_{n+2}t}w{e}^{-ibt}\right)\end{array}$ (4.6)

where ${\left|z_1\right|}^2+\cdots +{\left|z_{n+1}\right|}^2-{\left|w\right|}^2=0$ for $\left(z\mathrm{,}w\right)=\left(z_1\mathrm{,}\cdots \mathrm{,}{z}_{n+1}\mathrm{,}w\right)\in V_0$ . If X is the vector field induced by $\rho \left(S^1\right)$ at $\left(z\mathrm{,}w\right)$ , then it follows

$X=\left(i{a}_1{z}_1\mathrm{,}\cdots \mathrm{,}i{a}_{n+1}{z}_{n+1}\mathrm{,}i{a}_{n+2}w\right)-\left(z_{1}ib\mathrm{,}\cdots \mathrm{,}{z}_{n+1}ib\mathrm{,}wib\right)\mathrm{.}$ (4.7)

Proposition 7. If $\rho \mathrm{<mo>\; :\; </mo>}{S}^1\to T^{n+2}\cdot S^1$ is a faithful lightlike 1-parameter group, then it has either one of the forms:

$\begin{array}{l}\rho \left(t\right)=\left(e^{it}\mathrm{,}\cdots \mathrm{,}{e}^{it}\right)\le \left(\stackrel{n+2}{\stackrel{︷}{\text{Sp}\left(1\right)\times \cdots \times \text{Sp}\left(1\right)}}\right)\le \text{Sp}\left(n+\mathrm{1,1}\right)\cdot \left\{1\right\}\mathrm{,}\\ \rho \left(t\right)=\left(\mathrm{1,}\cdots \mathrm{,1}\right)\cdot e^{it}\le \left\{1\right\}\cdot \text{Sp}\left(1\right)\le \text{Sp}\left(n+\mathrm{1,1}\right)\cdot \text{Sp}\left(1\right)\mathrm{.}\end{array}$ (4.8)

Proof. Case (i) $b=0$ . $X=\left(i{a}_1{z}_1\mathrm{,}\cdots \mathrm{,}i{a}_{n+1}{z}_{n+1}\mathrm{,}i{a}_{n+2}w\right)$ from (4.7) so that $\langle X,X\rangle =a_1^2{\left|z_1\right|}^2+\cdots +a_{n+1}^2{\left|z_{n+1}\right|}^2-a_{n+2}^2{\left|w\right|}^2=\left(a_1^2-a_{n+2}^2\right){\left|z_1\right|}^2+\cdots +\left(a_{n+1}^2-a_{n+2}^2\right){\left|z_{n+1}\right|}^2$ . Since $\mathrm{Re}\langle X,X\rangle =0$ and we assume $a_i\ge 0$ , it follows

$a_1=a_{n+2},\cdots ,a_{n+1}=a_{n+2}.$

As $a_i$ ’s are relatively prime, this implies

$a_1=\cdots =a_{n+1}=a_{n+2}=1.$

As a consequence $\rho \left(t\right)=\left(e^{it},\cdots ,e^{it}\right)\le \text{Sp}\left(n+1,1\right)\cdot \left\{1\right\}$ . In this case note that $T_x\left(S^{4n+3}\times S^3\right)=\mathrm{Im}ℍy\oplus \mathrm{Im}ℍx\oplus {\left\{x,y\right\}}^{\perp }$ such that $\langle x\mathrm{,}y\rangle \in {ℝ}^{\mathrm{<mo>\; *\; </mo>}}$ .

Case (ii) $b=1$ . It follows from (4.7) that

$X=\left(i{a}_1{z}_1\mathrm{,}\cdots \mathrm{,}i{a}_{n+1}{z}_{n+1}\mathrm{,}i{a}_{n+2}w\right)-\left(z_{1}i\mathrm{,}\cdots \mathrm{,}{z}_{n+1}i\mathrm{,}wi\right)\mathrm{.}$

Put $Y=\left(i{a}_1{z}_1,\cdots ,i{a}_{n+1}{z}_{n+1},i{a}_{n+2}w\right)$ , $W=\left(z_{1}i,\cdots ,z_{n+1}i,wi\right)=xi$ such that $X=Y-W$ and $\langle W,W\rangle =\bar{i}\langle x,x\rangle i=0$ . Calculate

$\begin{array}{l}\langle Y,Y\rangle =a_1^2{\left|z_1\right|}^2+\cdots +a_{n+1}{\left|z_{n+1}\right|}^2-a_{n+2}^2{\left|w\right|}^2,\\ \langle Y,W\rangle =a_1{\bar{z}}_1\bar{i}{z}_{1}i+\cdots +a_{n+1}{\bar{z}}_{n+1}\bar{i}{z}_{n+1}i-a_{n+2}\bar{w}\bar{i}wi,\\ \mathrm{Re}\langle Y,W\rangle =a_1{\left|z_1\right|}^2+\cdots +a_{n+1}{\left|z_{n+1}\right|}^2-a_{n+2}\left|w\right|=\mathrm{Re}\langle W,Y\rangle .\end{array}$ (4.9)

This shows

$\begin{array}{l}\mathrm{Re}\langle X,X\rangle =\mathrm{Re}\langle Y-W,Y-W\rangle \\ =\mathrm{Re}\langle Y,Y\rangle -2\mathrm{Re}\langle Y,W\rangle +\mathrm{Re}\langle W,W\rangle =\mathrm{R}\langle Y,Y\rangle -2\mathrm{Re}\langle Y,W\rangle \\ =\left(a_1^2-2a_1\right){\left|z_1\right|}^2+\cdots +\left(a_{n+1}^2-2a_{n+1}\right){\left|z_{n+1}\right|}^2-\left(a_{n+2}^2-2a_{n+2}\right){\left|w\right|}^2\\ =\left(\left(a_1^2-2a_1\right)-\left(a_{n+2}^2-2a_{n+2}\right)\right){\left|z_1\right|}^2+\cdots +\left(\left(a_{n+1}^2-2a_{n+1}\right)-\left(a_{n+2}^2-2a_{n+2}\right)\right){\left|z_{n+1}\right|}^2\\ =\left({\left(a_1-1\right)}^2-{\left(a_{n+2}-1\right)}^2\right){\left|z_1\right|}^2+\cdots +\left({\left(a_{n+1}-1\right)}^2-{\left(a_{n+2}-1\right)}^2\right){\left|z_{n+1}\right|}^2.\end{array}$

Thus

${\left(a_1-1\right)}^2={\left(a_{n+2}-1\right)}^2,\cdots ,{\left(a_{n+1}-1\right)}^2={\left(a_{n+2}-1\right)}^2.$ (4.10)

On the other hand, we may assume in general

$\begin{array}{l}{a}_1=\cdots =a_k=0.\\ a_{k+1}-1\le 0,\cdots ,a_l-1\le 0.\\ a_{l+1}-1\ge 0,\cdots ,a_{n+1}-1\ge 0.\end{array}$

(ii-1). Suppose $a_{n+2}-1\ge 0$ . As $0<a_j\le 1$ for $k+1\le j\le l$ , it implies $a_{k+1}=\cdots =a_l=1$ . Since ${\left(a_{k+1}-1\right)}^2={\left(a_{n+2}-1\right)}^2$ from (4.10), it follows $a_{n+2}=1$ . Again from (4.10), ${\left(a_j-1\right)}^2=0$ and so $a_j=1$ $\left(l+1\le j\le n+1\right)$ . Note that $a_i\ne 0$ because ${\left(a_i-1\right)}^2={\left(a_{n+2}-1\right)}^2=0$ . Thus $a_1=a_2=\cdots =a_{n+2}=1$ . This implies $\rho \left(t\right)=\left(e^{it}\mathrm{,}\cdots \mathrm{,}{e}^{it}\right)\cdot e^{it}$ .

(ii-2). Suppose $a_{n+2}-1<0$ . In this case $a_{n+2}=0$ . By (4.10), it follows that ${}^{\forall }{a}_i\ne 0$ and $a_1=\cdots =a_l=1$ , $a_i=2$ $\left(l+1\le i\le n+1\right)$ . Thus $\rho \left(t\right)=\left(1,\cdots ,1,e^{i2t},\cdots ,e^{i2t},1\right)\cdot e^{it}$ . This contradicts that nonzero $a_i$ ’s $\left(1\le i\le n+1\right)$ are relatively prime.

(ii-3). Suppose $a_{n+2}-1<0$ and $a_1=a_2=\cdots =a_{n+1}=0$ . Again $a_{n+2}=0$ and so $\rho \left(t\right)=\left(\mathrm{1,}\cdots \mathrm{,1}\right)\cdot e^{it}$ .

To complete the proof of the proposition we prove the following. Put $x=\left(z,w\right)=\left(z_1,\cdots ,z_{n+1},w\right)\in S^{4n+3}\times S^3\subset V_0$ such that $\langle x,x\rangle =0$ .

Lemma 8. Case (ii-1) does not occur.

Proof. It follows from (4.7) that

$X=\left(i{z}_1,\cdots ,i{z}_{n+1},iw\right)-\left(z_{1}i,\cdots ,z_{n+1}i,wi\right)=ix-xi.$ (4.11)

Put $x\text{=}p\text{+}jq\text{(}p\text{,}q\in {\text{C}}^{\text{n+2}}\text{)}\text{.}$ Then $X=2kq.$ As $\langle X,X\rangle =0$ implies $\langle q,q\rangle =0$ . On the other hand, the equation

$0=\langle x,x\rangle =(\langle p,p\rangle +\langle q,q\rangle )-2j\langle \bar{p},q\rangle$

shows $\langle p\mathrm{,}p\rangle +\langle q\mathrm{,}q\rangle =0,\langle \bar{p}\mathrm{,}q\rangle =0.$ Note that if $S^{\text{2n+1}}\times S^{\text{1}}$ is the canonical subset in $S^{\text{4n+3}}\times S^3,$ then $\langle p\mathrm{,}p\rangle =0$ if and only if $p\in S^{\text{2n+1}}\times S^1.$ Since X is a nontrivial vector field on $S^{\text{4n+3}}\times S^3,$ there is a point x in the open subset $S=S^{\text{4n+3}}\times S^3\backslash S^{\text{2n+1}}\times S^1$ such that $\langle p\mathrm{,}p\rangle \ne 0$ and thus $\langle X,X\rangle \ne 0$ on S, which contradicts that X is a lightlike vector field.

4.5. Proof of Theorem B

Applying Proposition 7 to a lightlike group $S^3$ we obtain:

Corollary 9. Let $\rho \mathrm{<mo>\; :\; </mo>}{S}^3\to \text{O}\left(4n+\mathrm{4,4}\right)$ be a faithful representation which preserves the metric $\mathrm{Re}\langle ,\rangle$ on $V_0$ . If $\rho \left(S^3\right)$ is a lightlike group on $S^{4n+3}\times S^3$ , then either one of the following holds.

$\begin{array}{l}\rho \left(S^3\right)=\text{diag}\left(\text{Sp}\left(1\right)\times \cdots \times \text{Sp}\left(1\right)\right)\le \text{Sp}\left(n+\mathrm{1,1}\right)\cdot \left\{1\right\}\mathrm{,}\\ \rho \left(S^3\right)=\left\{1\right\}\cdot \text{Sp}\left(1\right)\le \text{Sp}\left(n+\mathrm{1,1}\right)\cdot \text{Sp}\left(1\right)\mathrm{.}\end{array}$ (4.13)

Let $\left(\text{diag}\left(\text{Sp}\left(1\right)\times \cdots \times \text{S}p\left(1\right)\right)\cdot \text{Sp}\left(1\right)\mathrm{,}{S}^{4n+3}\times S^3\right)$ be as in (4.13). If $f\mathrm{<mo>\; :\; </mo>}{S}^{4n+3}\times S^3\to S^{4n+3}\times S^3$ is a map defined by $f\left(\left(z_1\mathrm{,}\cdots \mathrm{,}{z}_{n+1}\mathrm{,}w\right)\right)=\left(\bar{w}{z}_1\mathrm{,}\cdots \mathrm{,}\bar{w}{z}_{n+1}\mathrm{,}\bar{w}\right)$ , then for $a\in \text{Sp}\left(1\right)$ , $b\in \text{Sp}\left(1\right)$ ,

$f\left(\left(a{z}_1\mathrm{,}\cdots \mathrm{,}a{z}_{n+1}\mathrm{,}aw\bar{b}\right)\right)=\left(b\bar{w}{z}_1\mathrm{,}\cdots \mathrm{,}b\bar{w}{z}_{n+1}\mathrm{,}b\bar{w}\bar{a}\right)\mathrm{.}$

So the equivariant diffeomorphism $f$ induces a quotient equivariant diffeomorphism

$\begin{array}{l}\hat{f}\mathrm{<mo>\; :\; </mo>}\left(\text{Sp}\left(1\right)\mathrm{,}{S}^{4n+3}\times S^3/\rho \left(S^3\right)\right)\to \left(\text{diag}\left(\text{Sp}\left(1\right)\times \cdots \times \text{Sp}\left(1\right)\right)\mathrm{,}{S}^{4n+3}\right)\mathrm{.}\hfill \end{array}$ (4.14)

We prove Theorem B of Introduction.

Proof. Suppose that the pseudo-Riemannian manifold $\left(M\times S^3\mathrm{,}g\right)$ is conformally flat. Let $\pi ={\pi }_1\left(M\right)$ be the fundamental group and $\tilde{M}$ the universal covering of M. By the developing argument (cf. [7] ), there is a developing pair:

$\left(\rho \mathrm{,}\text{Dev}\right)\mathrm{<mo>\; :\; </mo>}\left(\pi \times S^3\mathrm{,}\tilde{M}\times S^3\mathrm{,}\tilde{g}\right)\to \left(\text{O}\left(4n+\mathrm{4,4}\right)\mathrm{,}{S}^{4n+3}\times S^3\mathrm{,}{g}^0\right)$

where $\text{Dev}$ is a conformal immersion such that ${\text{Dev}}^{\mathrm{<mo>\; *\; </mo>}}{g}^0=u\tilde{g}$ for some positive function u on $\tilde{M}\times S^3$ and $\rho \mathrm{<mo>\; :\; </mo>}\pi \times S^3\to \text{O}\left(4n+\mathrm{4,4}\right)$ is a holonomy homomorphism for which $\text{Dev}$ is equivariant with respect to $\rho$ .

By Corollary 9, if $\rho \left(S^3\right)=\left\{1\right\}\cdot \text{Sp}\left(1\right)\le \text{Sp}\left(n+\mathrm{1,1}\right)\cdot \text{Sp}\left(1\right)$ , then the normalizer of $\text{Sp}\left(1\right)$ in $\text{O}\left(4n+\mathrm{4,4}\right)$ is isomorphic to $\text{Sp}\left(n+\mathrm{1,1}\right)\cdot \text{Sp}\left(1\right)$ . In particular, $\rho \left(\pi \times S^3\right)=\rho \left(\pi \right)\times \text{Sp}\left(1\right)\le \text{Sp}\left(n+\mathrm{1,1}\right)\cdot \text{Sp}\left(1\right)$ where $\rho \left(S^3\right)=\left\{1\right\}\cdot \text{Sp}\left(1\right)$ . We have the commutative diagram:

(4.15)

where $\rho \left(\pi \right)\le \text{PSp}\left(n+\mathrm{1,1}\right)$ and $\text{dev}$ is an immersion which is $\hat{\rho }$ - equivariant.

If $\rho \left(S^3\right)=\text{diag}\left(\text{Sp}\left(1\right)\times \cdots \times \text{Sp}\left(1\right)\right)\le \text{Sp}\left(n+\mathrm{1,1}\right)\cdot \left\{1\right\}$ from (4.13), then $\rho \left(\pi \times S^3\right)=\rho \left(S^3\right)\cdot \rho \left(\pi \right)\le \text{diag}\left(\text{Sp}\left(1\right)\times \cdots \times \text{Sp}\left(1\right)\right)\cdot \text{Sp}\left(1\right)$ . Composed $f$ with $\text{Dev}$ , we have an equivariant diffeomorphism $\hat{f}\circ \text{dev}\mathrm{<mo>\; :\; </mo>}\left(\pi \mathrm{,}\tilde{M}\right)\to \left(\rho \left(\pi \right)\mathrm{,}{S}^{4n+3}\right)$ where $\rho \left(\pi \right)\le \text{diag}\left(\text{Sp}\left(1\right)\times \cdots \times \text{Sp}\left(1\right)\right)\le \text{PSp}\left(n+\mathrm{1,1}\right)$ . In each case taking the developing map either $\text{dev}$ of (4.15) or $\hat{f}\circ \text{dev}$ , a quaternionic 3 CR-manifold M is spherical, i.e. uniformized with respect to $\left(\text{PSp}\left(n+\mathrm{1,1}\right)\mathrm{,}{S}^{4n+3}\right)$ .

Conversely recall $\left({\omega }^0,{\left\{J_{\alpha }^0\right\}}_{\alpha =1,2,3}\right)$ is the standard quaternionic 3 CR-structure on $S^{4n+3}$ equipped with the standard hypercomplex structure $Q^0={\left\{J_{\alpha }^0\right\}}_{\alpha =1,2,3}$ on ${\text{D}}^0$ . Suppose that $\left(\omega ,{\left\{J_{\alpha }\right\}}_{\alpha =1,2,3}\right)$ is a spherical quaternionic 3 CR-structure on M with a quaternionic structure Q, then there exists a developing map $\text{dev}\mathrm{<mo>\; :\; </mo>}\tilde{M}\to S^{4n+3}$ such that

${\text{dev}}^{\ast }{\omega }^0=\lambda \tilde{\omega }\bar{\lambda }$

for some $ℍ$ -valued function $\lambda$ on $\tilde{M}$ with a lift of quaternionic 3 CR-structure $\tilde{\omega }$ . In particular, ${\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}\text{D}={\text{D}}^0$ and ${\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}Q=Q^0$ .

Let $\tilde{g}$ be a pseudo-Riemannian metric on $\tilde{M}\times S^3$ for $\tilde{\omega }$ which is a lift of $g$ and $\omega$ to $\tilde{M}\times S^3$ respectively. Put ${\omega }^{\prime }={\text{dev}}^{\ast }{\omega }^0$ . Let $\lambda =\sqrt{u}a$ be a function for $u>0$ and $a\in \text{Sp}\left(1\right)$ such that

${\omega }^{\prime }=ua\tilde{\omega }\bar{a}.$

By the definition, recall $d{\omega }_{\beta }^0\left(J_{\gamma }^{0}V\mathrm{,}W\right)=d{\omega }_{\alpha }^0\left(V\mathrm{,}W\right)$ $\left({}^{\forall }V\mathrm{,}W\in {\text{D}}^0\right)$ . The induced quaternionic structure ${\left\{{J^{\prime }}_{\alpha }\right\}}_{\alpha =1,2,3}$ for ${\omega }^{\prime }={\text{dev}}^{\ast }{\omega }^0$ is obtained as $d\left({\text{dev}}^{\mathrm{<mo>\; *\; </mo>}}{\omega }_{\beta }^0\right)\left({J^{\prime }}_{\gamma }X\mathrm{,}Y\right)=d\left({\text{dev}}^{\mathrm{<mo>\; *\; </mo>}}{\omega }_{\alpha }^0\right)\left(X\mathrm{,}Y\right)$ . Since $d{\omega }_{\beta }^0\left({\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}{J^{\prime }}_{\gamma }X\mathrm{,}{\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}Y\right)=d{\omega }_{\alpha }^0\left({\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}X\mathrm{,}{\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}Y\right)$ , taking $V={\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}X$ , we obtain

${\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}{J^{\prime }}_{\gamma }X=J_{\gamma }^0{\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}X\left({}^{\forall }X\in \text{D}\right)\mathrm{.}$ (4.16)

As ${\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}Q=Q^0=\text{span}\left(J_{\alpha }^0\mathrm{,}\alpha =\mathrm{1,2,3}\right)$ , note that ${\left\{{J^{\prime }}_{\alpha }\right\}}_{\alpha =1,2,3}\in Q$ .

On the other hand, let $g^{\prime }$ be the pseudo-Riemannian metric on $\tilde{M}\times S^3$ for ${\omega }^{\prime }$ , it follows from Theorem 4

$g^{\prime }=u\tilde{g}.$ (4.17)

Take the above element $a\in S^3$ and let $\rho \mathrm{<mo>\; :\; </mo>}{S}^3\to S^3$ be a homomorphism defined by $\rho \left(s\right)=as\bar{a}$ $\left({}^{\forall }s\in S^3\right)$ . Define a map $\text{dev}\times \rho \mathrm{<mo>\; :\; </mo>}\tilde{M}\times S^3\to S^{4n+3}\times S^3$ which makes the diagram commutative. (Here p is the projection onto the left summand.)

(4.18)

where both $p_{*}:\left(\text{D},\left\{{\text{<msup> <mo> J′ </mo> </msup>}}_{\alpha }\right\}\right)\to \left(\text{D},\left\{{\text{<msup> <mo> J′ </mo> </msup>}}_{\alpha }\right\}\right)$ and $p_{*}:\left({\text{D}}^0,\left\{{\text{J}}_{\alpha }^0\right\}\right)\to \left({\text{D}}^0,\left\{{\text{J}}_{\alpha }^0\right\}\right)$ are isomorphisms such that

$p_{*}\circ {J^{\prime }}_{\alpha }={J^{\prime }}_{\alpha }\circ p_{*}\text{and}{p}_{*}\circ J_{\alpha }^0=J_{\alpha }^0\circ p_{*}\left(\alpha =1,2,3\right).$ (4.19)

Recall from (3.5) that $g^0={\sigma }^0\odot p^{\mathrm{<mo>\; *\; </mo>}}{\omega }^0+d{p}^{\mathrm{<mo>\; *\; </mo>}}{\omega }_{\alpha }^0\circ J_{\alpha }^0$ . (We write p more pre- cisely.) Consider the pull-back metric

$\begin{array}{c}{\left(\text{dev}\times \rho \right)}^{\mathrm{<mo>\; *\; </mo>}}{g}^0\left(X\mathrm{,}Y\right)={\sigma }^0\odot p^{\mathrm{<mo>\; *\; </mo>}}{\omega }^0\left({\left(\text{dev}\times \rho \right)}_{\mathrm{<mo>\; *\; </mo>}}X\mathrm{,}{\left(\text{dev}\times \rho \right)}_{\mathrm{<mo>\; *\; </mo>}}Y\right)\\ +d{p}^{\mathrm{<mo>\; *\; </mo>}}{\omega }_{\alpha }^0\left(J_{\alpha }^0{\left(\text{dev}\times \rho \right)}_{\mathrm{<mo>\; *\; </mo>}}X\mathrm{,}{\left(\text{dev}\times \rho \right)}_{\mathrm{<mo>\; *\; </mo>}}Y\right)\mathrm{.}\end{array}$ (4.20)

Calculate the first and the second summand of (4.20) respectively.

$\begin{array}{c}{\left(\text{dev}\times \rho \right)}^{\mathrm{<mo>\; *\; </mo>}}\left({\sigma }^0\odot p^{\mathrm{<mo>\; *\; </mo>}}{\omega }^0\right)={\left(\text{dev}\times \rho \right)}^{\mathrm{<mo>\; *\; </mo>}}{\sigma }^0\odot {\left(\text{dev}\times \rho \right)}^{\mathrm{<mo>\; *\; </mo>}}{p}^{\mathrm{<mo>\; *\; </mo>}}{\omega }^0\\ ={\rho }^{\mathrm{<mo>\; *\; </mo>}}{\text{dev}}^{\mathrm{<mo>\; *\; </mo>}}{\sigma }^0\odot p^{\mathrm{<mo>\; *\; </mo>}}{\text{dev}}^{\mathrm{<mo>\; *\; </mo>}}{\omega }^0\mathrm{.}\end{array}$ (4.21)

$\begin{array}{l}d{p}^{\mathrm{<mo>\; *\; </mo>}}{\omega }_{\alpha }^0\left(J_{\alpha }^0{\left(\text{dev}\times \rho \right)}_{\mathrm{<mo>\; *\; </mo>}}X\mathrm{,}{\left(\text{dev}\times \rho \right)}_{\mathrm{<mo>\; *\; </mo>}}Y\right)\\ =d{\omega }_{\alpha }^0\left(J_{\alpha }^0{p}_{\mathrm{<mo>\; *\; </mo>}}{\left(\text{dev}\times \rho \right)}_{\mathrm{<mo>\; *\; </mo>}}X\mathrm{,}{p}_{\mathrm{<mo>\; *\; </mo>}}{\left(\text{dev}\times \rho \right)}_{\mathrm{<mo>\; *\; </mo>}}Y\right)\\ =d{\omega }_{\alpha }^0\left(J_{\alpha }^0{\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}{p}_{\mathrm{<mo>\; *\; </mo>}}X\mathrm{,}{\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}{p}_{\mathrm{<mo>\; *\; </mo>}}Y\right)\end{array}$

$=d{\omega }_{\alpha }^0\left({\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}{J^{\prime }}_{\alpha }{p}_{\mathrm{<mo>\; *\; </mo>}}X\mathrm{,}{\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}{p}_{\mathrm{<mo>\; *\; </mo>}}Y\right)$

$=d{\omega }_{\alpha }^0\left({\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}{p}_{\mathrm{<mo>\; *\; </mo>}}{J^{\prime }}_{\alpha }X\mathrm{,}{\text{dev}}_{\mathrm{<mo>\; *\; </mo>}}{p}_{\mathrm{<mo>\; *\; </mo>}}Y\right)$

$=d{p}^{\mathrm{<mo>\; *\; </mo>}}{\text{dev}}^{\mathrm{<mo>\; *\; </mo>}}{\omega }_{\alpha }^0\left({J^{\prime }}_{\alpha }X\mathrm{,}Y\right)=d\left(p^{\mathrm{<mo>\; *\; </mo>}}{\text{dev}}^{\mathrm{<mo>\; *\; </mo>}}{\omega }_{\alpha }^0\right)\circ {J^{\prime }}_{\alpha }\left(X\mathrm{,}Y\right)\mathrm{.}$ (4.22)

Thus

${\left(\text{dev}\times \rho \right)}^{\mathrm{<mo>\; *\; </mo>}}{g}^0=R_{\bar{a}}^{\mathrm{<mo>\; *\; </mo>}}{\text{dev}}^{\mathrm{<mo>\; *\; </mo>}}{\sigma }^0\odot p^{\mathrm{<mo>\; *\; </mo>}}{\text{dev}}^{\mathrm{<mo>\; *\; </mo>}}{\omega }^0+d\left(p^{\mathrm{<mo>\; *\; </mo>}}{\text{dev}}^{\mathrm{<mo>\; *\; </mo>}}{\omega }_{\alpha }^0\right)\circ {J^{\prime }}_{\alpha }\mathrm{.}$

Then it follows by the construction of (3.5) that ${\left(\text{dev}\times \rho \right)}^{\mathrm{<mo>\; *\; </mo>}}{g}^0$ is the corresponding pseudo-Riemannian metric for ${\text{dev}}^{\mathrm{<mo>\; *\; </mo>}}{\omega }^0={\omega }^{\prime }$ and so ${\left(\text{dev}\times \rho \right)}^{\mathrm{<mo>\; *\; </mo>}}{g}^0=g^{\prime }=u\tilde{g}$ by (4.17). Therefore $\left(\tilde{M}\times S^3\mathrm{,}\tilde{g}\right)$ is conformally flat and so is $\left(M\times S^3\mathrm{,}g\right)$ .

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Biquard, O. (2001) Quaternionic Contact Structures, Quaternionic Structures in Mathematics and Physics, Rome, World Sci. 1999 Publishing, River Edge, NJ. 23-30.
[2]	Webster, S. (1977) On the Transformation Group of a Real Hypersurfaces. Transactions of the American Mathematical Society, 231, 179-190. https://doi.org/10.1090/S0002-9947-1977-0481085-5
[3]	Alekseevsky, D.V. and Kamishima, Y. (2008) Pseudo-Conformal Quaternionic CR Structure on -Dimensional Manifolds. Annali di Matematica Pura ed Applicata, 187, 487-529. https://doi.org/10.1007/s10231-007-0053-2
[4]	Kobayashi, S. (1970) Transformation Groups in Differential Geometry. Ergebnisse Math., 70.
[5]	Lee, J. (1986) The Fefferman Metric and Pseudo Hermitian Invariants. Transactions of the American Mathematical Society, 296, 411-429. https://doi.org/10.1090/S0002-9947-1986-0837820-2
[6]	Fefferman, C. (1976) Monge-Ampère Equations, the Bergman Kernel, and Geometry of Pseudo-Convex Domains. Annals of Mathematics, 103, 395-416. https://doi.org/10.2307/1970945
[7]	Kulkarni, R. (1978) On the Principle of Uniformization. Journal of Differential Geo-metry, 13, 109-138. https://doi.org/10.4310/jdg/1214434351