Abstract

Keywords

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

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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