# Almost CR manifolds with contracting CR automorphism

Jae-Cheon Joo  Department of Mathematics, King Fahd University of Petroleum
and Minerals, 31261 Dhahran, Saudi Arabia<EMAIL_ADDRESS>and  Kang-Hyurk
Lee  Department of Mathematics and Research Institute of Natural Science,
Gyeongsang National University, Jinju, Gyeongnam, 52828, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we deal with a strongly pseudoconvex almost CR manifold with a
CR contraction. We will prove that the stable manifold of the CR contaction is
CR equivalent to the Heisenberg group model.

Research of the second named author was supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea government (No.
NRF-2019R1F1A1060891).

##  1\. Introduction

The aim of this paper is characterizing strongly pseudoconvex almost CR
manifolds admitting a CR contraction, equivalently pseudo-Hermitian manifolds
with a contracting pseudoconformal automorphism.

In geometry, one of main questions is how large transformation group of
manifolds are and how to chacterize manifolds with large transformation group.
In conformal and CR geometry, there is an interesting history about this
question. Obata considered conformal structures of Riemannian geometry and he
succeeded in characterizing the sphere $S^{n}$ under the noncompact action of
conformal transformation group on compact Riemannian manifolds ([7, 8]), and
those results were followed by Webster in CR case ([12]). In 1972,
Alekseevskii claimed that a Riemannian manifold either compact or noncompact
with a nonproper action of conformal transformation is conformally equivalent
to either $S^{n}$ or $\mathbb{R}^{n}$ ([1]), however it turned out later that
his proof contains an error. Alekseevskii’s claim has been reproved by Schoen
in both conformal and CR cases in 1995 ([10]; cf. [2, 11]). His theorem is as
follows.

######  Theorem 1.1 (Schoen [10]).

  1. (1)

If the conformal automorphism group of a Riemannian manifold $M$ acts
nonproperly, then $M$ is conformally equivalent to either the unit sphere or
the Euclidean space.

  2. (2)

If a strongly pseudoconvex CR manifold $M$ whose CR automorphism group acts
nonproperly on $M$, then $M$ is CR equivalent to either the Heisenberg group
or the unit sphere in the complex Euclidean space

On the other hand, in complex geometry it has been also studied the
characterization of domains in $\mathbb{C}^{n}$ by noncompact automorphism
group action. Among many interesting results, one of the most famous works was
proved by Wong-Rosay ([13, 9]): _a strongly pseudoconvex bounded domain
$\Omega$ in $\mathbb{C}^{n}$ with a noncompact automorphism group is
biholomorphic to the unit ball $\mathbb{B}^{n}$ in $\mathbb{C}^{n}$_. Note
that a holomorphic automorphism group $\mathrm{Aut}(\Omega)$ of $\Omega$
extends to a CR transformation group on the boundary $\partial\Omega$ and the
noncompactness of $\mathrm{Aut}(\Omega)$ implies that $\mathrm{Aut}(\Omega)$
acts nonproperly on $\partial\Omega$. Since the the unit sphere $S^{2n-1}$ in
$\mathbb{C}^{n}$ is the boundary of the unit ball and the Heisenberg group is
the unit sphere without one point, $S^{2n-1}\setminus\\{p\\}$, the Wong-Rosay
theorem can be regarded as the counterpart of Schoen’s result on the CR
manifold.

The Wong-Rosay theorem was generalized by Gaussier-Sukhov [3] and the second
author [5, 6] in almost complex setting. Without the integrability of the
almost complex structure, the unit ball with the standard complex structure is
not a unique model. There are many strongly pseudconvex domains, called _model
strongly pseudconvex domains_ whose automorphism groups are noncompact and
almost complex structures are non-integrable. In [6], there are a full
classification of such domains and a description of their automorphism groups.

As a CR counterpart, it is also natural to ask if we can generalize Schoen’s
result in almost CR case. In the previous paper [4], we have proved that the
boundary of each model strongly pseudoconvex domain is a strongly pseudoconvex
almost CR manifold which has a group structure that is isomorphic to that of
standard Heisenberg group as groups. We call them generalized Heisenberg
groups. It also turned out that the CR automorphism group of a generalized
Heisenberg group acts nonproperly since it always admit a contracting CR
automorphism as the standard one. A generalized Heisenberg group is known to
be parametrized by a skew symmetric complex $n\times n$ matrix $P$ and we
denote it by $\mathcal{H}_{P}$ (see Section 2.3 for defintion).

In [4], we proved the following theorem which is a generalization of Schoen’s
theorem for low dimensional almost CR manifolds.

######  Theorem 1.2 (Theorem 1.2 in [4]).

Let $M$ be a manifold with strongly pseudoconvex almost CR structure of real
dimension $5$ or $7$. Suppose that the CR automorphism group acts nonproperly
on $M$. If $M$ is noncompact, then $M$ is CR equivalent to $\mathcal{H}_{P}$
for a skew symmetric complex matrix $P$. If $M$ is compact, then the almost CR
structure of $M$ is integrable and $M$ is CR equivalent to the unit sphere in
the complex Euclidean space.

The difficulty of the proof of this theorem in arbitrary dimension comes from
the construction of the Yamabe equation which plays a crucial role in the
proof of Theorem 1.2 as well as Schoen’s theorem [10].

In order to get a characterization of $\mathcal{H}_{P}$ in arbitrary dimension
by its automorphism action, we restrict our attention to the case of admitting
a contracting automorphism. Recall that the generalized Heisenberg groups
admit contracting CR automorphisms. The main result in this paper is as
follows.

######  Theorem 1.3.

Let $M$ be a strongly pseudoconvex almost CR manifold. Suppose $M$ admits a
contracting CR automorphism. Then the stable manifold $\mathcal{W}$ of the
contracting automorphism is CR equivalent to $\mathcal{H}_{P}$ for some
complex skew-symmetric matrix $P$.

The idea of the proof of Theorem 1.3 is contructing a special contact form
$\theta_{0}$ on $\mathcal{W}$ for which $(\mathcal{W},\theta_{0})$ is indeed
equivalent to $(\mathcal{H}_{P},\theta_{P})$ as pseudo-Hermitian manifolds,
where $\theta_{P}$ is the canonical contact form on $\mathcal{H}_{P}$ defined
by (2.9). The consturcution is achieved by a standard dynamical method in
Section 3.

Outline of the paper. In section 2, we will introduce basic notions for the
strongly pseudconvex almost CR manifold, the pseudo-Hermitian structure and
their equivalence problem as studied in [4]. Then the contacting CR
automorpism and the stable manifold will be discussed in Section 3. And also
in Section 3.2, we will show that there is a smooth canonical contact form of
the contracting CR automorphism whose pseudo-Hermitian structure is CR
equivalent to the Heisenberg model for the proof of Theorem 1.3 (see Section
4). We also characterize the ambient manifold as the standard sphere under
further assumption that the contracting automorphism has another fixed point
which is contracting for the inverse map in Theorem 4.2.

Convention. Throughout this paper, we assume that every structure is
$C^{\infty}$-smooth. The summation convention is always assumed. Greek indices
will be used to indicate coefficients of complex and real tensors,
respectively. We will take the bar on Greek indices to denote the complex
conjugation of the corresponding tensor coefficients:
$\overline{Z}_{\alpha}=Z_{\bar{\alpha}}$,
$\bar{\omega}^{\alpha}=\omega^{\bar{\alpha}}$,
${\overline{R}}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\lambda\bar{\mu}}={R}^{\phantom{\bar{\beta}}\bar{\alpha}}_{\bar{\beta}\phantom{\bar{\alpha}}\bar{\lambda}\mu}$.

##  2\. Strongly pseudoconvex almost CR manifolds

In this section, we review the pseudo-Hermitian structure of the strongly
pseudoconvex almost CR manifold and its equivalence problem as in [4].

###  2.1. The pseudo-Hermitian structure

By an _almost CR manifold_ , we mean a real $(2n+1)$-dimensional manifold $M$
with a $2n$-dimensional subbundle $H$ of $TM$ equipped with an almost complex
structure $J$, which is called the almost CR structure. Since
$J^{2}=-\mathrm{Id}$, the complexified bundle $\mathbb{C}\otimes H$ splits
into two eigensubbundles $H_{1,0}$ and $H_{0,1}$ of complex dimension $n$,
corresponding to eigenvalues $i$ and $-i$ of $J$, respectively:

$\displaystyle\mathbb{C}\times H=H_{1,0}\oplus H_{0,1}\;,$ $\displaystyle
H_{1,0}=\left\\{X-iJX:X\in H\right\\}\;,\quad H_{0,1}=\left\\{X+iJX:X\in
H\right\\}\;.$

A complex-valued vector field is of type $(1,0)$ or $(0,1)$ if it is a section
to $H_{1,0}$ or $H_{0,1}$, respectively. The CR structure $(H,J)$ is
_integrable_ if the space of $(1,0)$-vector fields, $\Gamma(H_{1,0})$ is
closed under the Lie bracket:
$[\Gamma(H_{1,0}),\Gamma(H_{1,0})]\subset\Gamma(H_{1,0})$. If the Lie bracket
of two $(1,0)$-vector fields is always a section to $\mathbb{C}\times
H=H_{1,0}\oplus H_{0,1}$, then we call the CR structure is _partially
integrable_.

A diffeomorphism $f:M\rightarrow M$ is called a _CR automorphism_ if
$f_{*}H_{1,0}=H_{1,0}$. The group of all CR automorphisms of $M$ will be
denoted by $\mathrm{Aut}_{\mathrm{CR}}(M)$. This is the topological group with
the compact-open topology.

An almost CR manifold is said to be _strongly pseudoconvex_ if $H$ is a
contact distribution and if

$d\theta(X,JX)>0$

for a contact form $\theta$ and for any nonzero vector $X\in H$. Then the pair
$(M,\theta)$ is called a _pseudo-Hermitian manifold_. Let
$(Z_{\alpha})=(Z_{1},\ldots,Z_{n})$ be a $(1,0)$-frame (a local frame of
$H_{1,0}$). A $\mathbb{C}^{n}$-valued $1$-form
$(\theta^{\alpha})=(\theta^{1},\ldots,\theta^{n})$ is called the _admissible
coframe_ for $\theta$ with respect to $(Z_{\alpha})$ if

$\theta^{\alpha}(Z_{\beta})=\delta^{\alpha}_{\beta},\quad\theta^{\alpha}(Z_{\bar{\beta}})=0,\quad\theta^{\alpha}(T)=0$

for $\alpha,\beta=1,...,n$, where $T$ is the _characteristic vector field_ of
$\theta$, that is, a unique real tangent vector field $\theta(T)\equiv 1$ and
$T\,\raisebox{1.72218pt}{$\lrcorner$}\;d\theta\equiv 0$. The admissibility of
$(\theta^{\alpha})$ is equivalent to that the differential of the contact form
$\theta$ can be written by

(2.1)
$d\theta=2ig_{\alpha\bar{\beta}}\,\theta^{\alpha}\wedge\theta^{\bar{\beta}}+p_{\alpha\beta}\,\theta^{\alpha}\wedge\theta^{\beta}+p_{\bar{\alpha}\bar{\beta}}\,\theta^{\bar{\alpha}}\wedge\theta^{\bar{\beta}}$

where $(g_{\alpha\bar{\beta}})$ is a positive-definite hermitian symmetric
matrix which is called the Levi form, and $(p_{\alpha\beta})$ is a skew-
symmetric matrix. Throughout this paper, $(g^{\alpha\bar{\beta}})$ stands for
the inverse matrix of the Levi form $(g_{\alpha\bar{\beta}})$. If
$(p_{\alpha\beta})$ vanishes identically, then the almost CR structure is
partially integrable; thus we call $(p_{\alpha\beta})$ the _non-partial-
integrability_ with respect to $\theta,\theta^{\alpha}$.

Let

$\tilde{\theta}=e^{2f}\theta$

be another contact form, where $f$ is a real-valued smooth function on $M$.

######  Proposition 2.1 ([4]).

Let $(\theta^{\alpha})$ be a local admissible coframe for $\theta$ with
respect to a $(1,0)$-frame $(Z_{\alpha})$. Then the $\mathbb{C}^{n}$-valued
$1$-form $(\tilde{\theta}^{\alpha})$ defined by

$\tilde{\theta}^{\alpha}=e^{f}(\theta^{\alpha}+v^{\alpha}\theta)\;,$

is an admissible coframe for $\tilde{\theta}$, where $v^{\alpha}$ is defined
by

(2.2) $f_{\alpha}=iv_{\alpha}+p_{\alpha\beta}\,v^{\beta}.$

where $f_{\alpha}=Z_{\alpha}f$ and
$v_{\alpha}=v^{\bar{\beta}}g_{\alpha\bar{\beta}}$. Moreover, if we denote the
Levi form and non-partial-integrbility for $\tilde{\theta}$ with respect to
$(\tilde{\theta}^{\alpha})$ by $(\tilde{g}_{\alpha\bar{\beta}})$ and
$(\tilde{p}_{\alpha\beta})$, then

$\tilde{g}_{\alpha\bar{\beta}}=g_{\alpha\bar{\beta}},\quad\tilde{p}_{\alpha\beta}=p_{\alpha\beta}.$

The last argument of the proposition means that Equation 2.1 for
$\tilde{\theta}=e^{2}f$ is of the form

$d\tilde{\theta}=2ig_{\alpha\bar{\beta}}\,\tilde{\theta}^{\alpha}\wedge\tilde{\theta}^{\bar{\beta}}+p_{\alpha\beta}\,\tilde{\theta}^{\alpha}\wedge\tilde{\theta}^{\beta}+p_{\bar{\alpha}\bar{\beta}}\,\tilde{\theta}^{\bar{\alpha}}\wedge\tilde{\theta}^{\bar{\beta}}\;.$

###  2.2. The pseudo-Hermitian strucrure equation

Let $(M,\theta)$ be a pseudo-Hermitian manifold and $(\theta^{\alpha})$ be an
admissible coframe. The pseudo-Hermitian connection form
$({\omega}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}})$ for the Hermitian
metric $(g_{\alpha\bar{\beta}})$ in (2.1) is defined as follows.

######  Proposition 2.2 ([4]).

The pseudo-Hermitian connection form
$({\omega}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}})$ is uniquely
determined by following equations:

(2.3) $\displaystyle
d\theta^{\alpha}=\theta^{\beta}\wedge{\omega}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}+{T}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma}\theta^{\beta}\wedge\theta^{\gamma}+{N}^{\phantom{\bar{\beta}}\alpha}_{\bar{\beta}\phantom{\alpha}\bar{\gamma}}\theta^{\bar{\beta}}\wedge\theta^{\bar{\gamma}}+{A}^{\phantom{}\alpha}_{\phantom{\alpha}\bar{\beta}}\theta\wedge\theta^{\bar{\beta}}+{B}^{\phantom{}\alpha}_{\phantom{\alpha}\beta}{}\theta\wedge\theta^{\beta}\;,$
(2.4) $\displaystyle
dg_{\alpha\bar{\beta}}=\omega_{\alpha\bar{\beta}}+\omega_{\bar{\beta}\alpha}\;,$
(2.5)
$\displaystyle{T}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma}=-{T}^{\phantom{\gamma}\alpha}_{\gamma\phantom{\alpha}\beta}\;,\quad{N}^{\phantom{\bar{\beta}}\alpha}_{\bar{\beta}\phantom{\alpha}\bar{\gamma}}=-{N}^{\phantom{\bar{\gamma}}\alpha}_{\bar{\gamma}\phantom{\alpha}\bar{\beta}}\;,\quad
B_{\alpha\bar{\beta}}=B_{\bar{\beta}\alpha}\;.$

Then the connection form
$({\omega}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}})$ gives the
covariant derivative to tensor fields by

(2.6) $\displaystyle\nabla Z_{\alpha}$
$\displaystyle={\omega}^{\phantom{\alpha}\beta}_{\alpha\phantom{\beta}}\otimes
Z_{\beta}\;,$ $\displaystyle\nabla Z_{\bar{\alpha}}$
$\displaystyle={\omega}^{\phantom{\bar{\alpha}}\bar{\beta}}_{\bar{\alpha}\phantom{\bar{\beta}}}\otimes
Z_{\bar{\beta}}\;,$ $\displaystyle\nabla\theta^{\alpha}$
$\displaystyle=-{\omega}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}\otimes\theta^{\beta}\;,$
$\displaystyle\nabla\theta^{\bar{\alpha}}$
$\displaystyle=-{\omega}^{\phantom{\bar{\beta}}\bar{\alpha}}_{\bar{\beta}\phantom{\bar{\alpha}}}\otimes\theta^{\bar{\beta}}\;.$

Equations (2.1) and (2.3) are called the structure equations, and (2.4) and
(2.5) are called the compatibility conditions. Tensor coefficients
$p_{\alpha\beta}$,
${T}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma}$,
${N}^{\phantom{\bar{\beta}}\alpha}_{\bar{\beta}\phantom{\alpha}\bar{\gamma}}$,
${A}^{\phantom{}\alpha}_{\phantom{\alpha}\bar{\beta}}$,
${B}^{\phantom{}\alpha}_{\phantom{\alpha}\beta}{}$ are called the torsion
coefficients.

We define the curvature $2$-form by

${\Theta}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}=d{\omega}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}-{\omega}^{\phantom{\beta}\gamma}_{\beta\phantom{\gamma}}\wedge{\omega}^{\phantom{\gamma}\alpha}_{\gamma\phantom{\alpha}}\;.$

The coefficients of pseudo-Hermitian curvature is defined by

${R}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma\bar{\sigma}}=2{\Theta}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}(Z_{\gamma},Z_{\bar{\sigma}})\;.$

Let $\tilde{\theta}=e^{2f}\theta$ be a pseudoconformal change of the pseudo-
Hermitian structure $(M,\theta)$ and let $(\tilde{\theta}^{\alpha})$ the
admissible coframe for $\tilde{\theta}$ defined as in Proposition 2.1.

######  Proposition 2.3 ([4]).

The coefficieints of torsion and curvature of $\theta$ and $\tilde{\theta}$
with respect to $(\theta^{\alpha})$ and $(\tilde{\theta}^{\alpha})$ are
related as follows.

$\displaystyle{\widetilde{T}}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma}$
$\displaystyle=e^{-f}\left({T}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma}+v^{\alpha}p_{\beta\gamma}-p_{\gamma\rho}v^{\rho}{\delta}^{\phantom{}\alpha}_{\phantom{\alpha}\beta}+p_{\beta\rho}v^{\rho}{\delta}^{\phantom{}\alpha}_{\phantom{\alpha}\gamma}\right)\;,$
$\displaystyle{\widetilde{N}}^{\phantom{\bar{\beta}}\alpha}_{\bar{\beta}\phantom{\alpha}\bar{\gamma}}$
$\displaystyle=e^{-f}\left({N}^{\phantom{\bar{\beta}}\alpha}_{\bar{\beta}\phantom{\alpha}\bar{\gamma}}+v^{\alpha}p_{\bar{\beta}\bar{\gamma}}\right)\;,$
$\displaystyle{\widetilde{A}}^{\phantom{}\alpha}_{\phantom{\alpha}\bar{\beta}}$
$\displaystyle=e^{-2f}\left({A}^{\phantom{}\alpha}_{\phantom{\alpha}\bar{\beta}}-g^{\alpha\bar{\gamma}}v_{\bar{\gamma};\beta}-2iv^{\alpha}v_{\bar{\beta}}-2v^{\bar{\gamma}}{N}^{\phantom{\bar{\gamma}}\alpha}_{\bar{\gamma}\phantom{\alpha}\bar{\beta}}-2v^{\alpha}v^{\bar{\gamma}}p_{\bar{\gamma}\bar{\beta}}\right)\;,$
$\displaystyle{\widetilde{B}}^{\phantom{}\alpha}_{\phantom{\alpha}\beta}$
$\displaystyle=e^{-2f}\Big{(}{B}^{\phantom{}\alpha}_{\phantom{\alpha}\beta}-\frac{1}{2}(g^{\alpha\bar{\gamma}}v_{\bar{\gamma};\beta}+v_{\beta;\bar{\gamma}}g^{\alpha\bar{\gamma}})+{\delta}^{\phantom{}\alpha}_{\phantom{\alpha}\beta}f_{0}$
$\displaystyle\qquad\qquad-v^{\alpha}v^{\gamma}p_{\gamma\beta}-v_{\beta}v^{\bar{\gamma}}{p}^{\phantom{\bar{\gamma}}\alpha}_{\bar{\gamma}\phantom{\alpha}}-v^{\gamma}{T}^{\phantom{\gamma}\alpha}_{\gamma\phantom{\alpha}\beta}-v^{\bar{\gamma}}{T}^{\phantom{\bar{\gamma}\beta}\alpha}_{\bar{\gamma}\beta\phantom{\alpha}}\Big{)}\;,$
$\displaystyle{\widetilde{R}}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\lambda\bar{\mu}}$
$\displaystyle=e^{-2f}\Big{(}{R}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\lambda\bar{\mu}}-{\delta}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}f_{\lambda\bar{\mu}}-{\delta}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}f_{\bar{\mu}\lambda}+2ig_{\beta\bar{\mu}}{v}^{\phantom{}\alpha}_{\phantom{\alpha};\lambda}-2iv_{\beta\bar{\mu}}{\delta}^{\phantom{}\alpha}_{\phantom{\alpha}\lambda}$
$\displaystyle\qquad+i(g^{\alpha\bar{\gamma}}v_{\bar{\gamma};\beta}+v_{\beta;\bar{\gamma}}g^{\alpha\bar{\gamma}})g_{\lambda\bar{\mu}}-4{\delta}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}g_{\lambda\bar{\mu}}v^{\gamma}v_{\gamma}-4g_{\beta\bar{\mu}}{\delta}^{\phantom{}\alpha}_{\phantom{\alpha}\lambda}v^{\gamma}v_{\gamma}$
$\displaystyle\qquad-2ip_{\beta\gamma}v^{\gamma}v^{\alpha}g_{\lambda\bar{\mu}}-2iv_{\beta}v^{\bar{\gamma}}{p}^{\phantom{\bar{\gamma}}\alpha}_{\bar{\gamma}\phantom{\alpha}}g_{\lambda\bar{\mu}}+2iv^{\gamma}{T}^{\phantom{\gamma}\alpha}_{\gamma\phantom{\alpha}\beta}g_{\lambda\bar{\mu}}-2iv^{\bar{\gamma}}{T}^{\phantom{\bar{\gamma}\beta}\alpha}_{\bar{\gamma}\beta\phantom{\alpha}}g_{\lambda\bar{\mu}}\Big{)}\;.$
$\displaystyle{\widetilde{R}}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma\bar{\sigma}}$
$\displaystyle=e^{-2f}\Big{(}{R}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma\bar{\sigma}}-{\delta}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}f_{\gamma\bar{\sigma}}-{\delta}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}f_{\bar{\sigma}\gamma}+2ig_{\beta\bar{\sigma}}{v}^{\phantom{}\alpha}_{\phantom{\alpha};\gamma}-2iv_{\beta\bar{\sigma}}{\delta}^{\phantom{}\alpha}_{\phantom{\alpha}\gamma}$
$\displaystyle\qquad+i(g^{\alpha\bar{\lambda}}v_{\bar{\lambda};\beta}+v_{\beta;\bar{\lambda}}g^{\alpha\bar{\lambda}})g_{\gamma\bar{\sigma}}-4{\delta}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}}g_{\gamma\bar{\sigma}}v^{\lambda}v_{\lambda}-4g_{\beta\bar{\sigma}}{\delta}^{\phantom{}\alpha}_{\phantom{\alpha}\gamma}v^{\lambda}v_{\lambda}$
$\displaystyle\qquad-2ip_{\beta\lambda}v^{\lambda}v^{\alpha}g_{\gamma\bar{\sigma}}-2iv_{\beta}v^{\bar{\lambda}}{p}^{\phantom{\bar{\lambda}}\alpha}_{\bar{\lambda}\phantom{\alpha}}g_{\gamma\bar{\sigma}}+2iv^{\lambda}{T}^{\phantom{\lambda}\alpha}_{\lambda\phantom{\alpha}\beta}g_{\gamma\bar{\sigma}}-2iv^{\bar{\lambda}}{T}^{\phantom{\bar{\lambda}\beta}\alpha}_{\bar{\lambda}\beta\phantom{\alpha}}g_{\gamma\bar{\sigma}}\Big{)}\;.$

Moreover, the coefficients of covariant derivatives of $p_{\alpha\beta}$
change as follows.

$\displaystyle\tilde{p}_{\alpha\beta;\gamma}$
$\displaystyle=e^{-f}\left(p_{\alpha\beta;\gamma}-2p_{\alpha\beta}f_{\gamma}-2ip_{\alpha\gamma}v_{\beta}-2ip_{\gamma\beta}v_{\alpha}\right)\;,$
$\displaystyle\tilde{p}_{\alpha\beta;\bar{\gamma}}$
$\displaystyle=e^{-f}\left(p_{\alpha\beta;\bar{\gamma}}+2p_{\alpha\beta}f_{\bar{\gamma}}-2ip_{\alpha\lambda}v^{\lambda}g_{\beta\bar{\gamma}}-2ip_{\lambda\beta}v^{\lambda}g_{\alpha\bar{\gamma}}\right)\;.$

Here $v_{\bar{\gamma};\beta}$, $v_{\beta;\bar{\gamma}}$
$p_{\alpha\beta;\gamma}$, $p_{\alpha\beta;\bar{\gamma}}$ are coefficients of
covariant derivatives of the tensors $(v_{\alpha})$, $(p_{\alpha\beta})$ as
defined in (2.6):

$\displaystyle v_{\bar{\gamma};\beta}$
$\displaystyle=Z_{\beta}v_{\bar{\gamma}}-{\omega}^{\phantom{\bar{\gamma}}\bar{\sigma}}_{\bar{\gamma}\phantom{\bar{\sigma}}}(Z_{\beta})v_{\bar{\sigma}}\;,$
$\displaystyle v_{\beta;\bar{\gamma}}$
$\displaystyle=Z_{\bar{\gamma}}v_{\beta}-{\omega}^{\phantom{\beta}\sigma}_{\beta\phantom{\sigma}}(Z_{\bar{\gamma}})v_{\sigma}\;,$
$\displaystyle p_{\alpha\beta;\gamma}$
$\displaystyle=Z_{\gamma}p_{\alpha\beta}-{\omega}^{\phantom{\alpha}\sigma}_{\alpha\phantom{\sigma}}(Z_{\gamma})p_{\sigma\beta}-{\omega}^{\phantom{\beta}\sigma}_{\beta\phantom{\sigma}}(Z_{\gamma})p_{\alpha\sigma}\;,$
$\displaystyle p_{\alpha\beta;\bar{\gamma}}$
$\displaystyle=Z_{\bar{\gamma}}p_{\alpha\beta}-{\omega}^{\phantom{\alpha}\sigma}_{\alpha\phantom{\sigma}}(Z_{\bar{\gamma}})p_{\sigma\beta}-{\omega}^{\phantom{\beta}\sigma}_{\beta\phantom{\sigma}}(Z_{\bar{\gamma}})p_{\alpha\sigma}\;.$

###  2.3. Generalized Heiseneberg groups

Let $(z,t)=(z^{1},...,z^{n},t)$ be the standard coordinates on
$\mathbb{C}^{n}\times\mathbb{R}$ and let $P=(P_{\alpha\beta})$ be a constant
complex skew-symmetric matrix of size $n\times n$. Let

(2.7) $Z_{\alpha}=\frac{\partial}{\partial
z^{\alpha}}+\left(iz^{\bar{\alpha}}+P_{\alpha\beta}z^{\beta}\right)\frac{\partial}{\partial
t}$

for $\alpha=1,...,n$. The almost CR structure whose $H_{1,0}$ space is spanned
by $\\{Z_{\alpha}\\}$ will be denoted by $J_{P}$. We define the generalized
Heisenberg group corresponding to $P$ by the almost CR manifold
$(\mathbb{C}^{n}\times\mathbb{R},J_{P})$ and will denote it by
$\mathcal{H}_{P}$. In case $P=0$, $\mathcal{H}_{0}$ is the classic Heisenberg
group. Generally, we have

######  Proposition 2.4 ([4]).

Let $*=*_{P}$ be the binary operation on $\mathcal{H}_{P}$ defined by

$(z,t)*(z^{\prime},t^{\prime})=\left(z+z^{\prime},t+t^{\prime}+2\mathrm{Im}\,(z^{\alpha}z^{\prime\bar{\alpha}})-2\mathrm{Re}\,(P_{\alpha\beta}z^{\alpha}z^{\prime\beta})\right).$

Then this operation makes $\mathcal{H}_{P}$ a Lie group and $Z_{\alpha}$
defined in (2.7) is left invariant under $*$. In particular, $\mathcal{H}_{P}$
is a homogeneous almost CR manifold.

Since the canonical $(1,0)$-vector fields $Z_{1},\ldots,Z_{n}$ of $J_{P}$
satisfy

(2.8)
$[Z_{\alpha},Z_{\bar{\beta}}]=-2i\delta_{\alpha\bar{\beta}}\frac{\partial}{\partial
t}\;,\quad[Z_{\alpha},Z_{\beta}]=-2P_{\alpha\beta}\frac{\partial}{\partial
t}\;,$

the CR structure of $\mathcal{H}_{P}$ is integrable if and only if $P=0$. Each
model $\mathcal{H}_{P}$ admits the _dilation_
$D_{\tau}(z,t)=(e^{\tau}z,e^{2\tau}t)$ as a CR automorphism. Therefore the
Heisenberg group model has a CR-contraction at the origin $0$ so at every
point of $\mathcal{H}_{P}$ by the homogeneity.

Let $\theta_{P}$ be a real $1$-form on $\mathcal{H}_{P}$ defined by

(2.9)
$\theta_{P}=dt+iz^{\alpha}dz^{\bar{\alpha}}-iz^{\bar{\alpha}}dz^{\alpha}+P_{\alpha\beta}z^{\alpha}dz^{\beta}+P_{\bar{\alpha}\bar{\beta}}z^{\bar{\alpha}}dz^{\bar{\beta}}.$

Note that $\theta_{P}(Z_{\alpha})=0$ for every $\alpha=1,...,n$. Moreover,

$d\theta_{P}=2idz^{\alpha}\wedge
dz^{\bar{\alpha}}+P_{\alpha\beta}dz^{\alpha}\wedge
dz^{\beta}+P_{\bar{\alpha}\bar{\beta}}dz^{\bar{\alpha}}\wedge
dz^{\bar{\beta}}.$

Therefore, we see that $\mathcal{H}_{P}$ is a strongly pseudoconvex almost CR
manifold and $(\theta^{\alpha}=dz^{\alpha})$ is an admissible coframe for
$\theta$ which is dual to $(Z_{\alpha})$. Since
$d\theta^{\alpha}=ddz^{\alpha}=0$, we see that $(\mathcal{H}_{P},\theta_{P})$
has a vanishing pseudo-Hermitian connection
$({\omega}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}})$. Moreover all
coefficients of torsion and curvature, except $p_{\alpha\beta}$, vanish
identically and $p_{\alpha\beta}\equiv P_{\alpha\beta}$. Conversely,

######  Proposition 2.5 ([4]).

Let $(M,\theta)$ be a pseudo-Hermitian manifold. If

$p_{\alpha\beta;\gamma}\equiv{T}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma}\equiv{N}^{\phantom{\bar{\beta}}\alpha}_{\bar{\beta}\phantom{\alpha}\bar{\gamma}}\equiv{A}^{\phantom{}\alpha}_{\phantom{\alpha}\bar{\beta}}\equiv{B}^{\phantom{}\alpha}_{\phantom{\alpha}\beta}\equiv{R}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\lambda\bar{\mu}}\equiv
0$

for some (and hence for all) admissible coframe, then $(M,\theta)$ is locally
equivalent to $(\mathcal{H}_{P},\theta_{P})$ as a pseudo-Hermitian manifold.

###  2.4. CR mappings and diffeomorphisms of Heisenberg models

As we mentioned, except the standard Heisenberg group $\mathcal{H}_{0}$, every
CR structure of $\mathcal{H}_{P}$ is non-integrable. The non-integrability of
the structure gives some restriction to the CR mappings. The following lemma
for this observation will be used in the proof of Theorem 4.2.

######  Lemma 2.6 (cf. Proposition 3.3 in [4]).

Let $\mathcal{H}_{P}$ and $\mathcal{H}_{P^{\prime}}$ be non-integrable
Heisenberg group models of the same dimension. If there is a local CR
diffeomorphism

$G(z,t)=(w,s)=(w^{1},\ldots,w^{n},s)$

from $\mathcal{H}_{P}$ to $\mathcal{H}_{P}^{\prime}$, then

  * $(1)$

each $w^{\lambda}$ ($\lambda=1,\ldots,n$) is indepedent of $t$-variable and
holomorphic in $z$-variables;

  * $(2)$

$s(t)=rt+c(z)$ for some constant $r$ and real-valued function $c$;

  * $(3)$

the constant $r$ is determined by

$r=\frac{1}{n}\sum_{\alpha,\beta=1}^{n}\left\lvert\frac{\partial
w^{\beta}}{\partial z^{\alpha}}\right\rvert^{2}\;.$

###### Proof.

We will use $(w,s)=(w^{1},\ldots,w^{n},s)$ as a standard coordinates of
$\mathcal{H}_{P^{\prime}}$ also. Let
$K_{\alpha}(z)=iz^{\bar{\alpha}}+P_{\alpha\beta}z^{\beta}$ and
$K^{\prime}_{\lambda}(w)=iw^{\bar{\lambda}}+P^{\prime}_{\lambda\mu}w^{\mu}$.
Then the canonical $(1,0)$-frames $(Z_{\alpha})$ and $(Z^{\prime}_{\lambda})$
of $\mathcal{H}_{P}$ and $\mathcal{H}_{P^{\prime}}$ can be determined by

$Z_{\alpha}=\frac{\partial}{\partial
z^{\alpha}}+K_{\alpha}\frac{\partial}{\partial t}\quad\text{and}\quad
Z^{\prime}_{\lambda}=\frac{\partial}{\partial
w^{\lambda}}+K^{\prime}_{\lambda}\frac{\partial}{\partial s}\;,$

respectively. Let us consider

$dF(Z_{\alpha})=(Z_{\alpha}w^{\lambda})\frac{\partial}{\partial
w^{\lambda}}+(Z_{\lambda}w^{\bar{\mu}})\frac{\partial}{\partial
w^{\bar{\mu}}}+(Z_{\alpha}s)\frac{\partial}{\partial s}\;.$

Since $F$ is a CR mapping, we have
$dF(Z_{\alpha})={a}^{\phantom{\alpha}\lambda}_{\alpha\phantom{\lambda}}Z^{\prime}_{\lambda}$
for some functions ${a}^{\phantom{\alpha}\lambda}_{\alpha\phantom{\lambda}}$.
Comparing two expressions of $dF(Z_{\alpha})$, we can conclude that
${a}^{\phantom{\alpha}\lambda}_{\alpha\phantom{\lambda}}=Z_{\alpha}w^{\lambda}$
so

(2.10) $dF(Z_{\alpha})=(Z_{\alpha}w^{\lambda})Z^{\prime}_{\lambda}\;,\quad
Z_{\alpha}w^{\bar{\mu}}=0\;,\quad
Z_{\alpha}s=(Z_{\alpha}w^{\lambda})K^{\prime}_{\lambda}\circ F\;.$

From the non-integrability of $\mathcal{H}_{P}$, we can choose $\alpha,\beta$
so that $P_{\alpha\beta}\neq 0$. Then we have

(2.11)
$[dF(Z_{\alpha}),dF(Z_{\beta})]=dF([Z_{\alpha},Z_{\beta}])=-2P_{\alpha\beta}dF\left(\partial/\partial
t\right)$

from (2.8). One can see that the local vector field

$[dF(Z_{\alpha}),dF(Z_{\beta})]=[(Z_{\alpha}w^{\lambda})Z^{\prime}_{\lambda},(Z_{\beta}w^{\mu})Z^{\prime}_{\mu}]$

on $\mathcal{H}_{P^{\prime}}$ has no terms in $\partial/\partial
w^{\bar{\mu}}$. From

$-2P_{\alpha\beta}dF\left(\partial/\partial
t\right)=-2P_{\alpha\beta}\left(\frac{\partial w^{\lambda}}{\partial
t}\frac{\partial}{\partial w^{\lambda}}+\frac{\partial w^{\bar{\mu}}}{\partial
t}\frac{\partial}{\partial w^{\bar{\mu}}}+\frac{\partial s}{\partial
t}\frac{\partial}{\partial s}\right)$

we have

$\frac{\partial w^{\bar{\mu}}}{\partial t}=0\quad\text{so}\quad\frac{\partial
w^{\mu}}{\partial t}=0$

for each $\mu=1,\ldots,n$. Simultaneously, we have

$Z_{\alpha}w^{\lambda}=\frac{\partial w^{\lambda}}{\partial
z^{\alpha}}\;,\quad Z_{\bar{\alpha}}w^{\lambda}=\frac{\partial
w^{\lambda}}{\partial z^{\bar{\alpha}}}=0$

from (2.10). This implies that each $w^{\lambda}$ is independent of
$t$-variable and holomorphic in $z$-variables. Indeed $w^{\alpha}$ is defined
on the open set $\mathcal{U}_{1}=\pi_{1}(\mathcal{U})$ in $\mathbb{C}^{n}$
where $\mathcal{U}$ is the domain of $F$ in $\mathbb{C}^{n}\times\mathbb{R}$
and $\pi_{1}:\mathbb{C}^{n}\times\mathbb{R}\to\mathbb{C}^{n}$ is the natural
projection. When we write

$[dF(Z_{\alpha}),dF(Z_{\beta})]=\left[\frac{\partial w^{\lambda}}{\partial
z^{\alpha}}Z^{\prime}_{\lambda},\frac{\partial w^{\mu}}{\partial
z^{\beta}}Z^{\prime}_{\mu}\right]={b}^{\phantom{\alpha\beta}\lambda}_{\alpha\beta\phantom{\lambda}}Z^{\prime}_{\lambda}-2P^{\prime}_{\lambda\mu}\frac{\partial
w^{\lambda}}{\partial z^{\alpha}}\frac{\partial w^{\mu}}{\partial
z^{\beta}}\frac{\partial}{\partial s}$

by some functions
${b}^{\phantom{\alpha\beta}\lambda}_{\alpha\beta\phantom{\lambda}}$, applying
$dF\left(\partial/\partial t\right)=(\partial s/\partial t)\partial/\partial
s$ to (2.11) we have that
${b}^{\phantom{\alpha\beta}\lambda}_{\alpha\beta\phantom{\lambda}}=0$ for any
$\lambda$ so that

$\frac{\partial s}{\partial
t}=\frac{P^{\prime}_{\lambda\mu}}{P_{\alpha\beta}}(\partial_{\alpha}w^{\lambda})(\partial_{\beta}w^{\mu})\;.$

Note that $P_{\alpha\beta}\neq 0$ by the choice of $\alpha,\beta$. This
implies that $\partial s/\partial t$ is independent of $t$-variable so
$s(z,t)=r(z)t+c(z)$ for some smooth real-valued functions $r=\partial
s/\partial t$ and $c$ defined on $\mathcal{U}_{1}$. The third equation of
(2.10) can be written as

$Z_{\alpha}s=\frac{\partial s}{\partial z^{\alpha}}+K_{\alpha}\frac{\partial
s}{\partial t}=\frac{\partial w^{\lambda}}{\partial
z^{\alpha}}K^{\prime}_{\lambda}\circ F$

Since $K_{\alpha}$, $\partial s/\partial t$, $\partial w^{\lambda}/\partial
z^{\alpha}$, $K^{\prime}_{\lambda}\circ F$ are all independent of
$t$-variable, so is $\partial s/\partial z^{\alpha}$. This means that
$\partial r/\partial z^{\alpha}=0$ for each $\alpha$ because $\partial
s/\partial z^{\alpha}=(\partial r/\partial z^{\alpha})t+\partial c/\partial
z^{\alpha}$. Hence the real-valued function $r$ is a constant:
$s(z,t)=rt+c(z)$.

From
$[dF(Z_{\alpha}),dF(Z_{\bar{\beta}})]=dF([Z_{\alpha},Z_{\bar{\beta}}])=-2i\delta_{\alpha\bar{\beta}}dF(\partial_{t})$,
one can easily get

$\delta_{\alpha\bar{\beta}}\frac{\partial t}{\partial
s}=\delta_{\lambda\bar{\mu}}\frac{\partial w^{\lambda}}{\partial
z^{\alpha}}\frac{\partial w^{\bar{\mu}}}{\partial
z^{\bar{\beta}}}\;,\quad\text{so}\quad n\frac{\partial t}{\partial
s}=\sum_{\alpha,\lambda}\left\lvert\frac{\partial w^{\lambda}}{\partial
z^{\alpha}}\right\rvert^{2}\;.$

This completes the proof. ∎

The standard Heisenberg group $\mathcal{H}_{0}$ is CR-equivalent to
$S^{2n+1}\setminus\\{p\\}$, the unit sphere minus one point. Therefore
$\mathcal{H}_{0}\setminus\\{0\\}$ is also equivalent to
$S^{2n+1}\setminus\\{p,-p\\}$. By the symmetry of $S^{2n+1}$,
$\mathcal{H}_{0}\setminus\\{0\\}$ admits a non-trivial involution CR
automorphism:

$(z,t)\mapsto\left(\frac{z}{-\left\lvert
z\right\rvert^{2}+it},\frac{-t}{\left\lvert
z\right\rvert^{4}+t^{2}}\right)\;.$

But for a non-integrable Heisenberg model $\mathcal{H}_{P}$, every CR
automorphism of $\mathcal{H}_{P}\setminus\\{0\\}$ is extended to a global CR
automorphism of $\mathcal{H}_{P}$ (Proposition 3.3 in [4]).

As an application of Lemma 2.6, we can describe the CR automorphism group of
the Heisenberg group model (see [4]).

######  Theorem 2.7.

Let $\mathcal{H}_{P}$ be a Heisenberg group model.

  * $(1)$

$\mathcal{H}_{P}$ is CR equivalent to a Heisenberg group model
$\mathcal{H}_{P^{\prime}}$ if and only if there is a unitary matrix
$U=({U}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}})\in\mathrm{U}(n)$ with
$P_{\alpha\beta}={U}^{\phantom{\alpha}\lambda}_{\alpha\phantom{\lambda}}P^{\prime}_{\lambda\mu}{U}^{\phantom{\beta}\mu}_{\beta\phantom{\mu}}$.

  * $(2)$

The isotropy group
$\mathrm{Aut}_{0}(\mathcal{H}_{P})=\left\\{F\in\mathrm{Aut}(\mathcal{H}_{P}):F(0)=0\right\\}$
can be composed by

$\mathrm{Aut}_{0}(\mathcal{H}_{P})=\mathrm{Aut}_{0}(\mathcal{H}_{P},\theta_{P})\oplus\\{D_{\tau}\\}$

where $\mathrm{Aut}_{0}(\mathcal{H}_{P},\theta_{P})$ is the pseudo-Hermitian
isotropy group.

  * $(3)$

If $P=0$ equivalently $J_{P}$ is integrable, the pseudo-Hermitian isotropy
group $\mathrm{Aut}_{0}(\mathcal{H}_{0},\theta_{0})$ is isomorphic to the
unitary group $\mathrm{U}(n)$. If $P\neq 0$, then

$\mathrm{Aut}_{0}(\mathcal{H}_{P},\theta_{P})\simeq\left\\{U\in\mathrm{U}(n):U^{t}PU=P\right\\}\;.$

More precisely, every element of
$\mathrm{Aut}_{0}(\mathcal{H}_{P},\theta_{P})$ of the form:

$(z,t)\to(U(z),t)\;.$

for some $U\in\mathrm{U}(n)$.

##  3\. Contracting CR automorphism

In this section, we introduce the CR contraction and its stable manifold and
show that there is a canonical contact form of the CR contaction.

###  3.1. The CR contractions and the stable manifolds

Let $(M,\theta)$ be a paeudo-Hermitian manifold. We say that a CR automorphism
$\varphi$ of $M$ is _(weakly) contracting_ at $o\in M$ if

$\varphi^{*}\theta|_{o}=\mu\,\theta|_{o}$

for some real $0<\mu<1$. This definition is independent of choice of contact
form. Let $(\theta^{\alpha})$ is an admissible coframe for $\theta$ satisfying

$g_{\alpha\bar{\beta}}(o)=\delta_{\alpha\bar{\beta}}\;.$

Since $\varphi$ is preserving the CR structure, we can write

(3.1)
$\varphi^{*}\theta^{\alpha}=a^{\alpha}_{\beta}\theta^{\beta}+c^{\alpha}\theta,$

for some functions $a^{\alpha}_{\beta}$ and $c^{\alpha}$. Then from (2.1), we
see that

$\sum_{\gamma}a^{\gamma}_{\alpha}a^{\bar{\gamma}}_{\bar{\beta}}=\mu\delta_{\alpha\bar{\beta}}\quad\text{at
$o$.}$

Therefore, the matrix $A=(a^{\alpha}_{\beta})$ is a normal operator (hence,
diagonalizable) and its eigenvalues have modulus $\sqrt{\mu}$ at $o$. By a
unitary change of the frame, we may assume

$A=(a^{\alpha}_{\beta})=\begin{pmatrix}\lambda_{1}&0&\cdots&0\\\
0&\lambda_{2}&\cdots&0\\\ \vdots&&\ddots&\vdots\\\
0&\cdots&0&\lambda_{n}\end{pmatrix}$

at $o$, where $|\lambda_{\alpha}|=\sqrt{\mu}$ for $\alpha=1,...,n$.

######  Proposition 3.1.

There exists a contact form $\theta$ such that $c^{\alpha}$ in (3.1) vanishes
at $o$.

###### Proof.

Let $\theta$ be any pseudo-Hermitian structure on $M$. Assume again that
$(\theta^{\alpha})$ be an admissible coframe for $\theta$ such that
$g_{\alpha\bar{\beta}}(o)=\delta_{\alpha\bar{\beta}}$ and
$A=(a^{\alpha}_{\beta})$ in (3.1) is diagonal at $o$. Let
$\tilde{\theta}=e^{2f}\theta$ be a pseudoconformal change of $\theta$, where
$f$ is a real-valued smooth function with $f(o)=0$ and let
$(\tilde{\theta}^{\alpha})$ be its admissible coframe defined by
$\tilde{\theta}^{\alpha}=e^{f}(\theta^{\alpha}+v^{\alpha}\theta)$ as in
Proposition 2.1. When we let
$\varphi^{*}\tilde{\theta}^{\alpha}=\tilde{a}_{\beta}^{\alpha}\tilde{\theta}^{\beta}+\tilde{c}^{\alpha}\tilde{\theta}$,
it follows that $\tilde{a}_{\beta}^{\alpha}\equiv a_{\beta}^{\alpha}$ and

$\tilde{c}^{\alpha}=c^{\alpha}-(a^{\alpha}_{\beta}-\mu\delta^{\alpha}_{\beta})v^{\beta}\quad\text{at
$o$.}$

We want to have

$\varphi^{*}\tilde{\theta}^{\alpha}=a^{\alpha}_{\beta}\tilde{\theta}^{\beta}\quad\text{at
$o$.}$

From (3.1) and Proposition 2.1, $v=(v^{\alpha}(o))$ must be a solution of

$(A-\mu I)v=c,$

where $c=(c^{\alpha}(o))$. Note that the matrix $A-\mu I$ is invertible at $o$
since the eigenvalues of $A$ have modulus $\sqrt{\mu}>\mu$. Therefore, the
above equation is uniquely solved for given $c$, and by taking $f$ whose
derivative $(f_{\alpha})$ at $o$ gives rise to the solution
$v=(v^{\alpha}(o))$ of (2.2), we see that $\tilde{\theta}$ satisfies the
condition of this proposition. ∎

From this proposition, we may assume that

(3.2) $\varphi^{*}|_{o}=\begin{pmatrix}\mu&0&\cdots&0\\\
0&\lambda_{1}&\cdots&0\\\ \vdots&&\ddots&\vdots\\\
0&\cdots&0&\lambda_{n}\end{pmatrix}$

with respect to $\theta,\theta^{\alpha}$.

Note that $TM=H\oplus\mathbb{R}T$. With regard to this decomposition, we can
define a Riemannian metric $ds^{2}$ on $M$ so that

$ds^{2}(T,T)=1,\quad ds^{2}(T,X)=0,\quad ds^{2}(X,Y)=d\theta(X,JY)$

for any $X,Y\in H$. For $x\in M$, we denote by $|x|$ the geodesic distance
from $o$ to $x$. Then (3.2) implies that

(3.3) $|\varphi(x)|\leq\eta|x|$

for some $0<\eta<1$ if $x$ is in a sufficiently small neighborhood $U$ of $o$.
Let $\mathcal{W}$ be the _stable manifold_ of $\varphi$. That is,
$x\in\mathcal{W}$ if and only if there exists a neighborhood $V$ of $x$ such
that $\varphi^{k}\rightarrow o$ uniformly on $V$ as $k\rightarrow\infty$. From
(3.3), it turns out $\mathcal{W}$ is a nonempty open subset of $M$. In fact,
it can be seen easily that

$\mathcal{W}=\bigcup_{k\in\mathbb{Z}}\varphi^{k}(U).$

###  3.2. The canonical contact form of the CR contraction

Let $M$ be a strongly pseudoconvex almost CR manifold and $\varphi$ be a CR
contraction at $o\in M$. We have $0<\mu<1$ with
$\varphi^{*}\theta|_{o}=\mu\theta|_{o}$ for any contact form $\theta$. If a
contact form $\theta$ on the stable manifold $\mathcal{W}$ satisfies

(3.4) $\varphi^{*}\theta=\mu\theta,$

we call $\theta$ a _canonical contact form_ of $\varphi$.

######  Proposition 3.2.

There exists a unique (up to constant multiple) continuous canonical contact
form $\theta$ on $\mathcal{W}$.

###### Proof.

We first show the uniqueness. Suppose that $\theta_{1}$ and $\theta_{2}$ are
two continuous contact form satisfying (3.4) and that
$\theta_{1}|_{o}=\theta_{2}|_{o}$. Then $\theta_{1}-\theta_{2}=u\theta_{1}$
for some continuous function $u$ on $\mathcal{W}$ with $u(o)=0$. Then for any
$x\in\mathcal{W}$,

$u(\varphi(x))\mu\theta_{1}|_{x}=\varphi^{*}(u\theta_{1})|_{x}=\varphi^{*}(\theta_{1}-\theta_{2})|_{x}=\mu(\theta_{1}-\theta_{2})|_{x}=u(x)\mu\theta_{1}|_{x}.$

Therefore, we have $u(\varphi(x))=u(x)$ for every $x\in\mathcal{W}$.
Conesequently,

$u(x)=u(\varphi^{k}(x))\rightarrow u(o)=0$

as $k\rightarrow\infty$ for every $x\in\mathcal{W}$.

Let $\tilde{\theta}$ be any smooth contact form on $M$. Let

$\theta_{k}=\frac{1}{\mu^{k}}\,(\varphi^{k})^{*}\tilde{\theta}$

for $k=1,2,...$. If we denote

$\theta_{k}=u_{k}\tilde{\theta},$

then it turns out that

$u_{k}(x)=\frac{1}{\mu^{k}}\,v(\varphi^{k-1}(x))\cdots
v(x)=\prod_{j=1}^{k}a_{j}(x),$

where $\varphi^{*}\tilde{\theta}=v\tilde{\theta}$ and
$a_{j}(x)=v(\varphi^{j-1}(x))/\mu$. Since $\varphi^{k}\rightarrow o$ locally
uniformly on $\mathcal{W}$, it suffices to show that $u_{k}$ converges
uniformly on $U$ to guarantee the convergence of $\theta_{k}$ on
$\mathcal{W}$. Note that the infintie product $\prod a_{j}(x)$ converges
absolutely and uniformly on $U$ if so is the infinite series
$\sum|a_{j}(x)-1|$. Since $0<v(x)\leq\mu+C|x|$ for some constant $C$ in $U$,
we see that

$0<a_{j}(x)\leq 1+C|\varphi^{j-1}(x)|\leq 1+C|x|\eta^{j-1}$

on $U$, for some constant $C>0$ by (3.3). Since $0<\eta<1$, we conclude that
$\sum_{j}|a_{j}-1|\leq C|x|\sum_{j}\eta^{j-1}$ converges uniformly on $U$.
This yields that $u_{k}\rightarrow u$ locally uniformly on $\mathcal{W}$ for
some positive continuous function $u$ on $\mathcal{W}$. It is obvious that
$\theta=u\tilde{\theta}$ satisfies (3.4). ∎

The next proposition implies that the canonical contact form $\theta$ in
Proposition 3.2 is smooth.

######  Proposition 3.3.

A canonical contact form $\theta$ induced by a contracting CR automorphism
$\varphi$ is indeed $C^{\infty}$-smooth on $\mathcal{W}$.

###### Proof.

We will prove for every positive integer $s$, $\\{D^{(s)}u_{k}:k\geq 0\\}$ is
locally uniformly bounded on $\mathcal{W}$, where $D^{(s)}$ represents the
$s$-th order differential operator. We assume $\tilde{\theta}$ and
$\theta^{\alpha}$ were chosen such that $\varphi^{*}$ has the form of (3.2) at
$o$. Let $T$ be the characteristic vector field for $\tilde{\theta}$ and
$Z_{\alpha}$ be the dual frame for $\\{\theta^{\alpha}\\}$. Then there exists
a local coordinates $(z^{1},...,z^{n},t)\in\mathbb{C}^{n}\times\mathbb{R}$
such that

$\left.\frac{\partial}{\partial
z^{\alpha}}\right|_{o}=Z_{\alpha}|_{o},\quad\left.\frac{\partial}{\partial
t}\right|_{o}=T|_{o}.$

Therefore, the Jacobian of $\varphi$ in this coordinate system has the same
form with (3.2) at $o$. Shrinking the neighborhood $U$ of $o$ if necessary, we
may assume

(3.5) $|D\varphi|\leq\eta$

on $U$ for some $0<\eta<1$, where $D$ is the differential operator in this
coordinate system.

Claim. For each positive integer $s$, there exists a polynomial $P_{s}$
depending on $s$ and $\|\varphi\|_{(s)}$ such that

$|D^{(s)}\varphi^{j}|\leq P_{s}(j)\eta^{j-s+1}$

on $U$ whenever $j\geq s$, where $\|\cdot\|_{(s)}$ denotes the
$C^{s}(U)$-norm.

Assume for a while that this claim is true. Since
$a_{j}=v(\varphi^{j-1})/\mu$, if $j\geq s$, then

(3.6)
$|D^{(s)}a_{j}|=\mu^{-1}|D^{(s)}(v(\varphi^{j-1}))|\leq\widetilde{P}_{s}(j)\eta^{j-s}$

on $U$ for some polynomial $\widetilde{P}_{s}$ depending on $\|v\|_{(s)}$ and
$P_{1},...,P_{s}$, from the chain rule and Claim.

Now, we finish the proof of this proposition. Recall that
$u_{k}=\prod_{j=1}^{k}a_{j}.$ Therefore,

$\displaystyle D^{(s)}u_{k}$ $\displaystyle=$
$\displaystyle\sum_{j}D^{(s)}a_{j}\prod_{l\neq j}a_{l}+\sum_{j_{1}\neq
j_{2}}D^{(s-1)}a_{j_{1}}Da_{j_{2}}\prod_{l\neq j_{1},l\neq j_{2}}a_{l}$
$\displaystyle+\cdots+{\sum_{j_{1},...,j_{s}}}^{\prime}Da_{j_{1}}\cdots
Da_{j_{s}}\prod_{l\neq j_{1},...,l\neq j_{s}}a_{l},$

where ${\sum}^{\prime}$ means the summation over mutually distinct indices.
Taking $U$ sufficiently small, we may assume $\mu/2\leq v(x)\leq 2\mu$ for
$x\in U$. Then we have $a_{j}\geq 1/2$ on $U$ for every $j$. Since the
infinite product $\prod_{j}a_{j}$ converges uniformly on $U$, we see that

$\prod_{l\neq j_{1},...,l\neq j_{s}}a_{l}=a_{j_{1}}\cdots
a_{j_{s}}\prod_{l}a_{l}\leq 2^{s}\prod_{l}a_{l}\leq C<\infty$

on $U$ for some constant $C>0$. Therefore from (3.6),

$|D^{(s)}u_{k}|\leq C\sum_{j}\widetilde{P}_{s}(j)\eta^{j-s}\leq
C^{\prime}<\infty$

on $U$ for some constants $C$ and $C^{\prime}$ independent of $k$. This
implies that $\\{D^{(s)}u_{k}:k\geq 1\\}$ is locally uniformly bounded on
$\mathcal{W}$. Therefore, for each $s>1$, we have a subsequence of
$\\{u_{k}\\}$ convergent to $u$ in local $C^{s-1}$-sense. Since $s$ is
arbitrary, we conclude that $u$ is $C^{\infty}$-smooth.

Now we prove the claim above. In case $s=1$, it turns out easily that

$|D\varphi^{j}|\leq\eta^{j}$

on $U$ for every $j\geq 1$, from the chain rule and (3.5). If $s=2$, then

$\displaystyle D^{(2)}\varphi^{j}$ $\displaystyle=$ $\displaystyle
D(D\varphi(\varphi^{j-1})\cdot D\varphi^{j-1})$ $\displaystyle=$
$\displaystyle D^{(2)}\varphi(\varphi^{j-1})\cdot D\varphi^{j-1}\cdot
D\varphi^{j-1}+D\varphi(\varphi^{j-1})\cdot D^{(2)}\varphi^{j-1}.$

Therefore,

(3.7)
$|D^{(2)}\varphi^{j}|\leq\|\varphi\|_{(2)}\eta^{2j-2}+\eta|D^{(2)}\varphi^{j-1}|\leq\|\varphi\|_{(2)}\eta^{j-1}+\eta|D^{(2)}\varphi^{j-1}|.$

for every $j\geq 2$. If $j=2$, then $|D^{(2)}\varphi^{2}|\leq
2\|\varphi\|_{(2)}\eta.$ Therefore, if we choose
$P_{2}(j)=\|\varphi\|_{(2)}j$, the recursive relation (3.7) implies that

$|D^{(2)}\varphi^{j}|\leq P_{2}(j)\eta^{j-1}$

on $U$ for every $j\geq 2$.

For more general $s>1$, if we have already chosen $P_{1}\equiv
1,P_{2},\ldots,P_{s-1}$, then we can also show that

(3.8) $|D^{(s)}\varphi^{j}|\leq Q(j)\eta^{j-s+1}+\eta|D^{(s)}\varphi^{j-1}|$

on $U$ for $j\geq s$, where $Q(j)$ is a polynomial in $j$ determined by
$\|\varphi\|_{(s)}$ and $P_{1},...,P_{s-1}$. Therefore, if $j=s$, then
$|D^{(s)}\varphi^{j}|\leq C_{s}\eta$ for some $C_{s}$ depending on
$\|\varphi_{s}\|$ and if we choose $P_{s}$ a polynomial with degree greater
than that of $Q$ such that

$P_{s}(s)\geq C_{s},\quad P_{s}(j)\geq Q(j)+P_{s}(j-1),$

then the relation (3.8) yields the conclusion. ∎

##  4\. Proofs of main theorems

In this section, we will prove Theorem 1.3 and characterize the ambient
manifold $M$ as the standard sphere under further assumption that the
contracting automorphism has another fixed point which is contracting for the
inverse map.

Proof of Theorem 1.3. Assume that $M$ is a strongly pseudoconvex almost CR
manifold and $\varphi$ is a contracting CR automorphism at $o\in M$. Let
$\theta$ be a canonical pseudo-Hermitian structure on $\mathcal{W}$ as
obtained in Section 3.2 so that

$\varphi^{*}\theta=\mu\theta$

on the stable manifold $\mathcal{W}$ for some constant $0<\mu<1$. We denote by
$\|\cdot\|_{\theta}$ the norm of tensors measured by $\theta$. For instance,
for
$T={T}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma}\theta^{\beta}\wedge\theta^{\gamma}\otimes
Z_{\alpha}$,

$\|T\|_{\theta}^{2}={T}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma}{T}^{\phantom{\bar{\rho}}\bar{\eta}}_{\bar{\rho}\phantom{\bar{\eta}}\bar{\sigma}}\,g_{\alpha\bar{\eta}}g^{\beta\bar{\rho}}g^{\gamma\bar{\sigma}}.$

Obviously, $\|\cdot\|_{\theta}$ does not depend on the choice of coframe
$\\{\theta^{\alpha}\\}$.

Let $U$ be a sufficiently small neighborhood of $o$. Since $\mu$ is constant,
$\varphi^{-k}(U)$ strictly increases as $k$ increases. Therefore, for a point
$x\in\mathcal{W}$, there exists $k_{0}\geq 1$ such that $x\in\varphi^{-k}(U)$
for every $k\geq k_{0}$. Let
$\theta_{k}=(\varphi^{-k})^{*}\theta=\mu^{-k}\theta$. Since

$\varphi^{k}:(\varphi^{-k}(U),\theta)\rightarrow(U,\theta_{k})$

is a pseudohermitain equivalence, we have

$\|T(x)\|_{\theta}=\|T_{k}(x_{k})\|_{\theta_{k}},$

where $x_{k}=\varphi^{k}(x)\in U$ and $T_{k}$ is the torsion tensor for
$\theta_{k}$ on $U$. Since $\theta_{k}=\mu^{-k}\theta$ a psudoconformal change
of $\theta$, Proposition 2.3 yields that

$\|T_{k}(x_{k})\|_{\theta_{k}}=\mu^{k/2}\|T(x_{k})\|_{\theta}.$

Therefore,

$\|T(x)\|_{\theta}=\mu^{k/2}\|T(x_{k})\|_{\theta}\leq\mu^{k/2}\sup_{y\in
U}\|T(y)\|_{\theta}\rightarrow 0$

as $k\rightarrow\infty$. This means that $T\equiv 0$ on $\mathcal{W}$.
Similarly, we can show that

${N}^{\phantom{\bar{\beta}}\alpha}_{\bar{\beta}\phantom{\alpha}\bar{\gamma}}={A}^{\phantom{}\alpha}_{\phantom{\alpha}\bar{\beta}}={B}^{\phantom{}\alpha}_{\phantom{\alpha}\beta}=p_{\alpha\beta;\gamma}={R}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\gamma\bar{\sigma}}\equiv
0$

on $\mathcal{W}$. Therefore, we can conclude that $(M,\theta)$ is locally
equivalent to $(\mathcal{H}_{P},\theta_{P})$ from Proposition 2.5. We may
assume $(U,\theta)$ is pseudo-Hermitian equivalent to $(V,\theta_{P})$ for
some neighborhood $V$ of $0$ in $\mathcal{H}_{P}$ and let

$F:(U,\theta)\rightarrow(V,\theta_{P})$

be a pseudo-Hermitian equivalence map such that $F(o)=0$. Let
$\Lambda:\mathcal{H}_{P}\rightarrow\mathcal{H}_{P}$ be the contracting CR
automorphism on $\mathcal{H}_{P}$ defined by

(4.1) $\Lambda(z,t)=(\sqrt{\mu}\,z,\mu t).$

Note that

$\Lambda^{*}\theta_{P}=\mu\theta_{P}.$

Therefore, the map

$F_{k}:=\Lambda^{-k}\circ
F\circ\varphi^{k}:\varphi^{-k}(U)\rightarrow\Lambda^{-k}(V)$

is a pseudo-Hermitian equivalence between $(\varphi^{-k}(U),\theta)$ and
$(\Lambda^{-k}(V),\theta_{P})$. Since $\\{F_{k}:k\geq 1\\}$ is not compactly
divergent, ($F^{k}(o)=0$ for all $k$) we conclude that $\\{F_{k}\\}$ has a
subsequence converging to a pseudo-Hermitian equivalence
$\widetilde{F}:(\mathcal{W},\theta)\rightarrow(\mathcal{H}_{P},\theta_{P})$.
Altogether, we have proved Theorem 1.3. ∎

######  Remark 4.1.

Unlike Theorem 1.2, it is not very clear how to characterize the ambient
manifold $M$ of arbitrary dimension. According to former results in [10] and
[4], it may be natural to expect either $M=\mathcal{W}\simeq\mathcal{H}_{P}$
for some $P$ in case $M$ is noncompact, or $M$ is CR equivalent with the
standard sphere and
$\mathcal{W}=M\setminus\\{\mbox{pt}\\}\simeq\mathcal{H}_{0}$ in case $M$ is
compact. In [10, 4], this global characterization could be done from the
derivative estimates obtained from a PDE theory on the CR or subconformal
Yamabe equations (see Proposition 2.1 and 2.1’ in [10]). On the other hand, it
is still unknown whether there exists a curvature invariant which satisfies a
Yamabe-type equation under pseudoconformal changes or not, if $M$ is a
strongly pseudoconvex almost CR manifold of arbitrary dimension.

If $M$ is the standard sphere and if $\varphi$ is a contracting CR
automorphism with a contracting fixed point $o$, then $\varphi^{-1}$ is also a
contracting CR automorphism with another contracting fixed point $o^{\prime}$.
The next theorem states that the converse is also true even in almost CR
cases.

######  Theorem 4.2.

Let $M$ be a strongly pseudoconvex almost CR manifold and let $\varphi$ be a
contracting CR automorphism with a contracting fixed point $o$. Let
$\mathcal{W}$ be the stable manifold of $\varphi$ with respect to $o$. Suppose
that $\mathcal{W}\neq M$ and that there is a contracting fixed point
$o^{\prime}\in\partial\mathcal{W}$ of $\varphi^{-1}$. Then $M$ is CR
equivalent to the standard sphere $S^{2n+1}\subset\mathbb{C}^{n+1}$.

###### Proof.

Let $\widetilde{F}:(\mathcal{W},\theta)\to(\mathcal{H}_{P},\theta_{P})$ be a
pseudo-Hermitian equivalence for the canonical contact form $\theta$ of $o$.
Applying Theorem 1.3, we have also a pseudo-Hermitian equivalence
$\widetilde{F}^{\prime}:(\mathcal{W}^{\prime},\theta^{\prime})\rightarrow(\mathcal{H}_{P^{\prime}},\theta_{P^{\prime}})$
where $\mathcal{W}^{\prime}$ is the stable neighborhood of $o^{\prime}$ with
respect to $\varphi^{-1}$ and $\theta^{\prime}$ is the canonical contact form
on $\mathcal{W}^{\prime}$ with

$(\varphi^{-1})^{*}\theta^{\prime}=\mu^{\prime}\theta^{\prime}$

for some $0<\mu^{\prime}<1$. Let
$\mathcal{V}=\mathcal{W}\cap\mathcal{W}^{\prime}$ which is an open subset of
$M$ admitting $o,o^{\prime}$ as boundary points. Take a point
$p\in\mathcal{V}$. Then

(4.2) $\varphi^{k}(p)\to o\quad\text{and}\quad\varphi^{-k}(p)\to o^{\prime}$

as $k\to\infty$. Let us consider

  1. (1)

$\mathcal{U}=\widetilde{F}(\mathcal{V})$,
$\mathcal{U}^{\prime}=\widetilde{F}^{\prime}(\mathcal{V})$ open subsets of
$\mathcal{H}_{P}$ and $\mathcal{H}_{P^{\prime}}$, respectively

  2. (2)

$(z_{0},t_{0})=\widetilde{F}(p)\in\mathcal{U}$,
$(w_{0},s_{0})=\widetilde{F}^{\prime}(p)\in\mathcal{U}^{\prime}$,

  3. (3)

$\psi=\widetilde{F}\circ\varphi\circ\widetilde{F}^{-1}$,
$\psi^{\prime}=\widetilde{F}^{\prime}\circ\varphi\circ\widetilde{F}^{\prime-1}$
corresponding CR automorphisms of $\varphi$ in $\mathcal{H}_{P}$ and
$\mathcal{H}_{P^{\prime}}$, respectively.

Since $\mathcal{U}$ and $\mathcal{U}^{\prime}$ are open in
$\mathbb{C}^{n}\times\mathbb{R}$, we may assume that $z_{0}\neq 0$ and
$w_{0}\neq 0$. Theorem 2.7 implies that

$\psi(z,t)=(\mu^{1/2}U(z),\mu
t)\;,\quad\psi^{\prime}(w,s)=(\mu^{\prime-1/2}U^{\prime}(w),\mu^{\prime-1}s)$

for some $U,U^{\prime}\in\mathrm{U}(n)$. Since $z_{0},w_{0}\neq 0$, we have

$z_{k}=\mu^{k/2}U^{k}(z_{0})\to 0\quad\text{and}\quad
w_{k}=\mu^{\prime-k/2}U^{k}(w_{0})\to\infty$

in $\mathbb{C}^{n}$.

Let us consider the local CR diffeomorphism
$G=\widetilde{F}^{\prime}\circ\widetilde{F}^{-1}:\mathcal{U}\subset\mathcal{H}_{P}\to\mathcal{U}^{\prime}\subset\mathcal{H}_{P^{\prime}}$
and denote by

$G(z,t)=(w,s)=(w^{1},\ldots,w^{n},s)\;.$

Now suppose that $\mathcal{H}_{P}$ is non-integrable and let $\mathcal{U}_{1}$
be the projection image of $\mathcal{U}$ to $\mathbb{C}^{n}$. Then
$w=(w^{1},\ldots,w^{n})$ is independent of $t$ and is a holomorphic mapping
from $\mathcal{U}_{1}$ to $\mathbb{C}^{n}$ by (1) of Lemma 2.6. Moreover

$\displaystyle G\circ\psi^{k}$
$\displaystyle=(\widetilde{F}^{\prime}\circ\widetilde{F}^{-1})\circ(\widetilde{F}\circ\varphi^{k}\circ\widetilde{F}^{-1})$
$\displaystyle=(\widetilde{F}^{\prime}\circ\varphi^{k}\circ\widetilde{F}^{\prime-1})\circ(\widetilde{F}^{\prime}\circ\widetilde{F}^{-1})=\psi^{\prime
k}\circ G$

and

$\displaystyle G(\psi^{k}(z_{0},t_{0}))$
$\displaystyle=\big{(}w(\psi^{k}(z_{0},t_{0})),s(\psi^{k}(z_{0},t_{0}))\big{)}=\big{(}w(z_{k}),s(\psi^{k}(z_{0},t_{0}))\big{)}\;,$
$\displaystyle\psi^{\prime k}(G(z_{0},t_{0})$ $\displaystyle=\psi^{\prime
k}(w_{0},s_{0})=(\mu^{\prime-k/2}U^{\prime k}(w_{0}),\mu^{\prime-k}s_{0})\;.$

Therefore we have $w(z_{k})=w_{k}=\mu^{\prime-k/2}U^{-k}(w_{0})$. As a
conclusion, the holomorphic mapping $w:\mathcal{U}_{1}\to\mathbb{C}^{n}$ is
not defined and diverges to infinity at the origin $0$ of $\mathbb{C}^{n}$,
namely,

$z_{k}\to 0\quad\text{but}\quad w(z_{k})=w_{k}\to\infty$

in $\mathbb{C}^{n}$. This implies that

$\sum_{\alpha,\beta=1}^{n}\left\lvert\frac{\partial w^{\beta}}{\partial
z^{\alpha}}(z_{k})\right\rvert^{2}\to\infty\quad\text{as $k\to\infty$.}$

It is a contradiction to Statement (3) of Lemma 2.6. This implies
$\mathcal{H}_{P}$ is integrable. Similarly $\mathcal{H}_{P^{\prime}}$ is also
an integrable Heisenberg group and as a consequence,
$\mathcal{W}\cup\mathcal{W}^{\prime}$ is an open submanifold of $M$ whose CR
structure is integrable. Notice that the automorphism $\varphi$ acts on
$\mathcal{W}\cup\mathcal{W}^{\prime}$ and generates a noncompact orbit with
two fixed points $o$ and $o^{\prime}$. Therefore
$\mathcal{W}\cup\mathcal{W}^{\prime}$ must be CR equivalent to the standard
sphere by Schoen’s theorem [10]. Then the conclusion follows since
$\mathcal{W}\cup\mathcal{W}^{\prime}$ is an open submanifold of $M$ without
boundary and hence $M=\mathcal{W}\cup\mathcal{W}^{\prime}$. ∎

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