# On the automorphism group of parabolic structures and closed aspherical
manifolds

Oliver Baues  Department of Mathematics
University of Fribourg
Chemin du Musée 23
CH-1700 Fribourg, Switzerland<EMAIL_ADDRESS>and  Yoshinobu Kamishima
Department of Mathematics, Josai University
Keyaki-dai 1-1, Sakado, Saitama 350-0295, Japan<EMAIL_ADDRESS>
(Date: September 22, 2023)

###### Abstract.

In this expository paper we discuss several properties on closed aspherical
parabolic ${{\mathsf{G}}}$-manifolds $X/\Gamma$. These are manifolds
$X/\Gamma$, where $X$ is a smooth contractible manifold with a parabolic
${{\mathsf{G}}}$-structure for which $\Gamma\leq\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X)$ is a discrete subgroup acting properly
discontinuously on $X$ with compact quotient. By a parabolic
${\mathsf{G}}$-structure on $X$ we have in mind a Cartan structure which is
modeled on one of the classical parabolic geometries arising from simple Lie
groups ${\mathsf{G}}$ of rank one. Our results concern in particular the
properties of the automorphism groups $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)$. Our main results show that the
existence of certain parabolic ${{\mathsf{G}}}$-structures can pose strong
restrictions on the topology of compact aspherical manifolds $X/\Gamma$ and
their parabolic automorphism groups. In this realm we prove that any compact
aspherical standard $CR$-manifold with virtually solvable fundamental group is
diffeomorphic to a quotient of a Heisenberg manifold of complex type with its
standard $CR$-structure. Furthermore we discuss the analogue properties of
standard quaternionic contact manifolds in relation to the quaternionic
Heisenberg group.

###### Key words and phrases:

$CR$-structure, Pseudo-Hermitian structure, Conformal structure, Quaternionic
contact structure, Sasaki metric, (Hyper-) Kähler manifold

###### 2010 Mathematics Subject Classification:

22E41, 53C10, 57S20, 53C55

This work was partially supported by JSPS grant No 22K03319

##  1\. Introduction

Let ${{\mathbb{H}}}^{n+1}_{{\mathbb{K}}}$ be the ${\mathbb{K}}$-hyperbolic
space over ${\mathbb{K}}={\mathbb{R}},{\mathbb{C}}$ or ${\mathbb{H}}$.
(Compare [14] and the definitions therein.) The group of
${\mathbb{K}}$-hyperbolic isometries $\mathop{\rm
PU}\nolimits(n+1,1;{\mathbb{K}})$ of ${{\mathbb{H}}}^{n+1}_{{\mathbb{K}}}$
extends to an analytic action on the boundary sphere
$\partial{{\mathbb{H}}}^{n+1}_{{\mathbb{K}}}=S^{|{\mathbb{K}}|(n+1)-1}$. This
corresponds to the rank one standard parabolic geometry on
$S^{|{\mathbb{K}}|(n+1)-1}=\mathop{\rm
PU}\nolimits(n+1,1;{\mathbb{K}})/P_{\mathbb{K}}$ where $P_{\mathbb{K}}$ is a
maximal parabolic subgroup. In terms of classical geometry, according to
whether ${\mathbb{K}}={\mathbb{R}},\,{\mathbb{C}}$ or ${\mathbb{H}}$, the
model spaces $S^{|{\mathbb{K}}|(n+1)-1}=S^{n},S^{2n+1}$ or $S^{4n+3}$ admit a
_conformally flat structure, spherical $CR$-structure_ or _spherical
quaternionic contact structure_ , respectively. (For precise definition of
these notions, see [6] or [12], and Section 3 of this article, for example.)

The geometry of these spaces is intricately reflected in the standard
graduation of the Lie algebra of $\mathop{\rm
PU}\nolimits(n+1,1;{\mathbb{K}})$, which is $\mathop{\rm PO}\nolimits(n+1,1)$
for ${\mathbb{K}}={\mathbb{R}}$ or $\mathop{\rm PU}\nolimits(n+1,1)$,
$\mathop{\rm PSp}\nolimits(n+1,1)$ for
${\mathbb{K}}={\mathbb{C}},{\mathbb{H}}$, where:

$\begin{split}\mathfrak{so}(n+1,1)&=\mathfrak{g}^{-1}+\mathfrak{g}^{0}+\mathfrak{g}^{1}={\mathbb{R}}^{n}+(\mathfrak{so}(n)+{\mathbb{R}})+({\mathbb{R}}^{n})^{*},\\\
\mathfrak{su}(n+1,1)&=\mathfrak{g}^{-2}+\mathfrak{g}^{-1}+\mathfrak{g}^{0}+\mathfrak{g}^{1}+\mathfrak{g}^{2}\\\
&={\rm
Im}{\mathbb{C}}+{\mathbb{C}}^{n}+(\mathfrak{u}(n)+{\mathbb{R}})+({\mathbb{C}}^{n})^{*}+({\rm
Im}{\mathbb{C}})^{*},\\\
\mathfrak{sp}(n+1,1)&=\mathfrak{g}^{-2}+\mathfrak{g}^{-1}+\mathfrak{g}^{0}+\mathfrak{g}^{1}+\mathfrak{g}^{2}\\\
&={\rm
Im}{\mathbb{H}}+{\mathbb{H}}^{n}+(\mathfrak{sp}(n)+\mathfrak{sp}(1)+{\mathbb{R}})+({\mathbb{H}}^{n})^{*}+({\rm
Im}{\mathbb{H}})^{*}\end{split}$

in which $\displaystyle P_{\mathbb{K}}$ is generated by the parabolic
subalgebra $\displaystyle\mathfrak{p}=\mathfrak{g}^{0}+\mathfrak{g}^{1}$ for
${\mathbb{K}}={\mathbb{R}}$, or
$\displaystyle\mathfrak{g}^{0}+\mathfrak{g}^{1}+\mathfrak{g}^{2}$ for
${\mathbb{C}},{\mathbb{H}}$, respectively (see [2]).

Put ${\mathsf{G}}=P_{\mathbb{K}}$, for brevity. By a _parabolic
${{\mathsf{G}}}$-structure_ on a manifold $M$, we thus mean either one of a
positive definite conformal structure, a strictly pseudo-convex
$CR$-structure, or a positive definite quaternionic contact structure
($qc$-structure for short) on $M$. If $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ is the group of structure preserving
transformations on such a parabolic ${{\mathsf{G}}}$-manifold $M$,
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ is called: (1) The group of
conformal transformations $\mathop{\rm Conf}\nolimits(M,[g])$. (2) The group
of $CR$-transformations $\mathop{\rm
Aut}\nolimits_{CR}(M,\\{{\mathsf{D}},J\\})$, or (3) The group of
$qc$-transformations $(\mathop{\rm
Aut}\nolimits_{qc}(M,{\mathsf{D}},\,\\{J_{\alpha}\\}_{\alpha=1}^{3})$,
respectively.

##### Rigidity of parabolic ${\mathsf{G}}$-manifolds with non-proper
automorphism group

An important observation on parabolic ${\mathsf{G}}$-manifolds is that the
automorphism group $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ does not
necessarily act properly on $M$. In particular this is the case for the model
spheres $S^{|{\mathbb{K}}|(n+1)-1}$. Given a parabolic
${{\mathsf{G}}}$-manifold $M$, in case $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ is a non-proper group the parabolic
${\mathsf{G}}$-manifold $M$ is completely determined by works of D. V.
Alekseevsky [2], J. Ferrand [18], R. Schoen [29], J. Lee [26], C. Frances
[19], S. Ivanov and D. Vassilev [21], Webster [30] and others, as follows:

######  Theorem A.

If the automorphism group $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ of a
parabolic ${\mathsf{G}}$-manifold $M$ does _not act properly_ , then $M$ with
its parabolic structure admits a structure preserving diffeomorphism to one of
the standard model spaces as specified in $(1),(2),(3):$ $(1)$ $M$ is
conformal to either the standard sphere $S^{n}$ or the euclidean space
${\mathbb{R}}^{n}$. Here it occurs

$\left(\mathop{\rm Iso}\nolimits(M),{\rm
Conf}(M)\right)=\begin{cases}\left({\rm O}(n+1),{\rm
PO}(n+1,1)\right)&(M=S^{n})\,,\\\ \left({\mathbb{R}}^{n}\rtimes{\rm
O}(n),{\mathbb{R}}^{n}\rtimes({\rm
O}(n)\times{\mathbb{R}}^{+})\right)&(M={\mathbb{R}}^{n})\,.\end{cases}$

$(2)$ $M$ has a spherical $CR$-structure isomorphic to either the standard
sphere $S^{2n+1}$ or the Heisenberg Lie group ${\mathcal{N}}$ (with its
canonical $CR$-structure). It occurs

$\left({\operatorname*{Psh}\,}_{CR}(M),\mathop{\rm
Aut}\nolimits_{CR}(M)\right)=\begin{cases}\left({\rm U}(n+1),{\rm
PU}(n+1,1)\right)\ \ \ \ (M=S^{2n+1}),\\\ \left({\mathcal{N}}\rtimes{\rm
U}(n),{\mathcal{N}}\rtimes({\rm U}(n)\times{\mathbb{R}}^{+})\right)\
(M={\mathcal{N}}).\,\end{cases}$

$(3)$ $M$ has a spherical $qc$-structure isomorphic to either the standard
sphere $S^{4n+3}$ or the quaternionic Heisenberg nilpotent Lie group
${\mathcal{M}}$.
$({\operatorname*{Psh}\,}_{qc}(M),\mathop{\rm Aut}\nolimits_{qc}(M))$ occurs

$\begin{cases}\left(\mathop{\rm Sp}\nolimits(n+1)\cdot\mathop{\rm
Sp}\nolimits(1),\ {\rm PSp}(n+1,1),\ S^{4n+3}\right)\ \ \ \ \ \ \ \ \ \ \
(M=S^{4n+3}),\,\\\ \left({\mathcal{M}}\rtimes{\rm Sp}(n)\cdot\mathop{\rm
Sp}\nolimits(1),\ {\mathcal{M}}\rtimes({\rm Sp}(n)\cdot\mathop{\rm
Sp}\nolimits(1)\times{\mathbb{R}}^{+}),\ {\mathcal{M}}\,\right)\
(M={\mathcal{M}}).\,\\\ \end{cases}$

The theorem also gives the pairs
$({\operatorname*{Psh}\,}_{{\mathsf{G}}}(M),\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M))$ for the respective cases (1), (2), (3). Here
${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$ is the maximal subgroup of
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ that is acting properly on $M$.
In particular, in case (1), ${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$ is the
subgroup of $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M))$ which preserves the
canonical Riemannian metric that defines the conformal structure on $M$. In
cases (2) and (3), the group ${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$
coincides with the subgroup of $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$
that preserves the canonical structure defining contact forms.

#####  ${\mathsf{G}}$-Hermitian subgroups of $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$

As the rigidity theorem shows, for any parabolic ${\mathsf{G}}$-manifold $M$,
there exists a maximal subgroup ${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$ of
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ that is acting properly on $M$.
If the parabolic structure on $M$ is of type (2) or (3), it is defined by the
conformal class of a contact form $\omega$. In this case we define
${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M,\omega)$ to be the subgroup of
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ that preserves $\omega$. Note
that ${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M,\omega)$ always acts properly
on $M$ (see [6]), that is, ${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M,\omega)$
is contained in ${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$. The groups
${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M,\omega)$ are also called
${\mathsf{G}}$-Hermitian subgroups of $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$, see Section 3 below. As to the precise
relation of ${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$ with the
${\mathsf{G}}$-Hermitian subgroups we additionally have the following:

######  Theorem 1 (see Proposition 3.1, [6]).

Let $M$ be a parabolic ${\mathsf{G}}$-manifold and let $H\leq\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ be a closed subgroup. Then there exists a
canonical cohomology class $[\lambda_{{\mathsf{G}}}]$ in the differentiable
cohomology group $H^{1}_{d}(H,C^{\infty}(M,{\mathbb{R}}^{+}))$ with the
following properties:

  1. i)

If $[\lambda_{{\mathsf{G}}}]=0$ and the ${\mathsf{G}}$-structure is conformal,
then there exists a Riemannian metric $g$ representing the conformal structure
such that $H$ is contained in $\mathop{\rm Iso}\nolimits(M,g)$.

  2. ii)

If $[\lambda_{{\mathsf{G}}}]=0$ and the parabolic structures is of type $(2)$
or $(3)$, there exists a compatible contact form $\omega$ (that is, $\omega$
is representing the parabolic structure) such that $H$ is contained in
$\displaystyle{\operatorname*{Psh}\,}_{{\mathsf{G}}}(M,\omega)$.

  3. iii)

The group $H$ acts properly on $M$ if and only if
$[\lambda_{{\mathsf{G}}}]=0$.

If $M$ is compact, Theorem 1 combined with Theorem A implies that $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ is a compact Lie group, except if $M$ is one
of the standard parabolic ${\mathsf{G}}$-spheres as described in Theorem A.

In the following we will be in particular interested in compact parabolic
${\mathsf{G}}$-manifolds.

##### Automorphisms of aspherical parabolic ${\mathsf{G}}$-manifolds

A compact manifold $M$ is called aspherical if its universal covering manifold
$X$ is contractible. In that case $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ is compact and its identity component
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)^{0}$ is a compact torus. The
latter fact is a consequence of the following fundamental result on compact
Lie group actions on closed aspherical manifolds:

######  Theorem B (Conner and Raymond [15, 16]).

Let $X/\Gamma$ be a closed aspherical Riemannian manifold. Then the isometry
group $\mathop{\rm Iso}\nolimits(X/\Gamma)$ is a finite group or the identity
component $\mathop{\rm Iso}\nolimits(X/\Gamma)^{0}$ is isomorphic to a
$k$-torus $T^{k}$. Moreover, there is a central group extension(3.2),
$1{\rightarrow}{\mathbb{Z}}^{k}{\rightarrow}\,\Gamma{\longrightarrow}\,Q{\rightarrow}1$,
where $Q=\Gamma/{\mathbb{Z}}^{k}$.

Theorem B shows that if the fundamental group $\Gamma$ of $M$ has no normal
solvable subgroup then $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ is a
finite group. Well known examples of such aspherical manifolds $M$ are compact
locally symmetric spaces of non-compact type (without local flat factors). In
this context, we remark the following general fact:

######  Theorem 2.

Let $X/\Gamma$ be a closed aspherical manifold such that $\Gamma$ has no
normal solvable subgroup. If $X/\Gamma$ admits a parabolic
${{\mathsf{G}}}$-structure, then its automorphism $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)$ is a finite group which is isomorphic
to a subgroup of $\mathop{\rm Out}\nolimits(\Gamma)$. In particular, $\Gamma$
is of finite index in its normalizer $N_{\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X)}(\Gamma)$.

In the theorem $\mathop{\rm Out}\nolimits(\Gamma)=\mathop{\rm
Aut}\nolimits(\Gamma)/\mathop{\rm Inn}\nolimits(\Gamma)$ denotes the outer
automorphism group of $\Gamma$.

#####  $CR$\- and $qc$-structures on closed aspherical manifolds

From our viewpoint of _parabolic ${{\mathsf{G}}}$_-structures, we are
interested how the existence of a parabolic ${{\mathsf{G}}}$-structure
determines the topology of $X/\Gamma$ and in particular its smooth structure.
We show that the existence of certain $CR$\- or $qc$-parabolic structures on
$X/\Gamma$, that admit a non-trivial connected group of automorphisms
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)^{0}$, poses a strong
restriction on $\Gamma$. In fact, we show that under the assumption that
$\Gamma$ is virtually solvable any standard $CR$-manifold $X/\Gamma$ is
_diffeomorphic_ to a Heisenberg type manifold which is derived from the
standard model space in (2) of Theorem A. In the $qc$-case a much stronger
rigidity result holds, namely we show that _any_ closed aspherical standard
$qc$-manifold $X/\Gamma$ is _$qc$ -equivalent_ to a $qc$-manifold of
quaternionic Heisenberg type (as in (3) of Theorem A).

##### Heisenberg manifolds

Recall from Theorem A that the Heisenberg Lie group ${\mathcal{N}}$ admits a
maximal proper subgroup of affine transformations ${\mathcal{N}}\rtimes{\rm
U}(n)$. The group ${\mathcal{N}}\rtimes{\rm
U}(n)={\operatorname*{Psh}\,}_{CR}({\mathcal{N}})$ then preserves the
canonical left-invariant $CR$-structure, and the canonical pseudo-Hermitian
structure on the Heisenberg group ${\mathcal{N}}$ (see for example, [22, 7]).
Given a torsion-free discrete uniform subgroup $\Gamma$ contained in
${\mathcal{N}}\rtimes{\rm U}(n)$, the compact quotient manifold

$M={\mathcal{N}}/\Gamma$

is then called a _Heisenberg infra-nilmanifold_.

##### Standard $CR$-structures

Suppose that $M$ admits a $CR$-structure with contact form $\omega$, where $M$
is a $2n+1$-dimensional manifold. Then the $CR$-structure is called _standard_
if the Reeb vector field associated to $\omega$ generates a one-parameter
subgroup of ${\operatorname*{Psh}\,}_{CR}(M,\omega)$.

In that sense, every Heisenberg infra-nilmanifold carries a canonical standard
$CR$-structure, where $\mathop{\rm
Aut}\nolimits_{CR}(M)^{0}=\operatorname*{Psh}\,_{CR}(M,\omega)^{0}=S^{1}$.
(The structure is induced from the standard $CR$-structure on the Heisenberg
group ${\mathcal{N}}$.) See Section 4 for details.

Further examples of (aspherical) standard $CR$-manifolds may be constructed as
$S^{1}$-bundles over any compact Hermitian locally symmetric spaces $B$, using
the Kähler class of $B$ to determine the circle bundle. (In fact, this
construction works over every compact Kähler manifold $B$. See [7] and the
references therein).

##### Virtually solvable fundamental group $\Gamma$

Let $M=X/\Gamma$ be a closed aspherical manifold such that $\Gamma$ is a
virtually solvable group (which means that $\Gamma$ contains a solvable
subgroup of finite index). In fact, since $M$ is aspherical, $\Gamma$ is a
torsion-free virtually polycyclic group. It is known that every such group
occurs as the fundamental group of a compact aspherical manifold $X/\Gamma$.
Note further that the fundamental group $\Gamma$ determines $M$ up to
homeomorphism, but not necessarily up to diffeomorphism unless some further
geometric structure on $M$ is specified that enforces smooth rigidity (cf. [4]
and the references therein.)

Every _standard_ $CR$-manifold with solvable fundamental group turns out to be
diffeomorphic to a Heisenberg infra-nilmanifold:

######  Theorem 3.

Let $M=X/\pi$ be a $2n+1$-dimensional closed aspherical positive definite
strictly pseudoconvex _standard_ $CR$-manifold. If $\pi$ is virtually solvable
then then there exists a discrete faithful representation
$\rho:\pi{\rightarrow}\,{\mathcal{N}}\rtimes{\rm U}(n)$ and $M$ is
diffeomorphic to the Heisenberg infra-nilmanifold ${\mathcal{N}}/\rho(\pi)$.
In particular, $\pi$ is virtually nilpotent and $\mathop{\rm
Aut}\nolimits_{CR}(M)^{0}=S^{1}$.

By choosing a representative contact form $\omega$ for the $CR$-structure on
$M$, we obtain a pseudo-Hermitian manifold $(M,(\omega,J))$. Note that a
standard pseudo-Hermitian manifold $(M,(\omega,J))$ is equivalent to a Sasaki
manifold $(M,g,(\omega,J),\xi)$ by assigning a positive definite Riemannian
metric $g=\omega\cdot\omega+d\omega\circ J$, called Sasaki metric. (Compare
[7, 10].)

By the existence of the $S^{1}$-action generated by the Reeb field, we see
that $M$ admits a fibering over a Kähler orbifold. We will prove:

Theorem 3′. If the fundamental group of a closed aspherical Sasaki manifold
$M$ is virtually solvable, then $M$ is diffeomorphic to a Heisenberg infra-
nilmanifold.

A Sasaki manifold is called _regular_ if the $S^{1}$-action generated by the
Reeb field is free. For a regular Sasaki manifold Theorem $3^{\prime}$ is
stated and proved in [7, Corollary 2, Proposition 6.10]. Assuming that the
fundamental group of $M$ is nilpotent, a proof of Theorem 3′ involving methods
of rational homotopy theory is provided in [28].

The following result is obtained by O. Baues and V. Cortés [5] which is used
for our proof.

######  Theorem C.

Let $X/\Gamma$ be a closed aspherical Kähler manifold with $\Gamma$ virtually
solvable. Then a finite cover of $X/\Gamma$ is biholomorphic to a complex
torus.

We next seek whether similar results hold for $4n+3$-dimensional
$qc$-manifolds.

##### Quaternionic Heisenberg manifolds

A quaternionic Heisenberg Lie group is a $4n+3$-dimensional nilpotent Lie
group $\mathcal{M}$ with center ${\mathbb{R}}^{3}$ whose quotient is
isomorphic to the quaternionic vector space ${\mathbb{H}}^{n}$, whose
structure determines the Lie product on ${\mathcal{M}}$ (see [2, 23]). The
group ${\mathcal{M}}$ admits a maximal proper subgroup of affine
transformations
${\operatorname*{Psh}\,}_{qc}({\mathcal{M}})={\mathcal{M}}\rtimes\mathop{\rm
Sp}\nolimits(n)\cdot\mathop{\rm Sp}\nolimits(1)$. Then
${\operatorname*{Psh}\,}_{qc}({\mathcal{M}})$ preserves the canonical
$qc$-structure and the canonical $qc$-Hermitian structure on ${\mathcal{M}}$.
(Compare Theorem A.) A quotient ${\mathcal{M}}/\Gamma$ by a torsion-free
discrete uniform subgroup
$\Gamma\leq{\operatorname*{Psh}\,}_{qc}({\mathcal{M}})$ is called a
quaternionic Heisenberg infra-nilmanifold.

For any $qc$-manifold $M$, let ${\mathcal{T}}=\\{\xi_{1},\xi_{2},\xi_{3}\\}$
denote the three-dimensional integrable distribution complementary to the
codimension three subbundle ${\mathsf{D}}$ on $M$ determined by the
$qc$-structure, called _$qck$ -distribution_ (see Section 3). If
${\mathcal{T}}$ generates a subgroup of ${\operatorname*{Psh}\,}_{qc}(M)$,
then $M$ is called a _standard_ $qc$-manifold.

##### Remark

When $M$ is a closed aspherical standard $qc$-manifold, ${\mathcal{T}}$
generates a three-torus $T^{3}\leq{\operatorname*{Psh}\,}_{qc}(M)$. In this
case, the $qc$-structure on $M$ thus does not give rise to a $3$-Sasaki
structure since the $qck$-distribution ${\mathcal{T}}$ does not generate an
$\mathop{\rm Sp}\nolimits(1)$-action on $M$ but a $T^{3}$-action [23, Theorem
5.4]. Then the quotient space $M/{\mathcal{T}}$ inherits a hyper-Kähler
structure from $M$.

We have the following strong rigidity property for standard $qc$-structures on
closed aspherical manifolds.

######  Theorem 4.

Let $X/\pi$ be a positive definite closed aspherical _standard_ $qc$-manifold.
Then $X/\pi$ is $qc$-isometric to a quaternionic Heisenberg infra-nilmanifold
${\mathcal{M}}/\pi$ where $\pi\leq{\mathcal{M}}\rtimes\mathop{\rm
PSp}\nolimits(n)$, is a discrete uniform subgroup. In particular, $\pi$ is
virtually nilpotent and $\mathop{\rm Aut}\nolimits_{qc}(X/\pi)^{0}=T^{3}$ is a
three-torus.

In the context of hyper-Kähler manifolds the following striking fact plays an
important role in the proof of Theorem 4:

##### Rigidity of aspherical hyper-Kähler manifolds

Recall that the quaternionic torus $T^{n}_{\mathbb{H}}$ admits a natural flat
homogeneous hyper-Kähler structure which is induced by a linear quaternionic
hermitian form on ${\mathbb{H}}^{n}$ (cf. [12]). We then have: Lemma D. Let
$M$ be a closed aspherical hyper-Kähler manifold. Then a finite cover of $M$
is _hypercomplexally isometric_ to a quaternionic torus $T^{n}_{\mathbb{H}}$
with its natural flat hyper-Kähler structure.  Here a hypercomplex isomorphism
is simultaneously a holomorphic diffeomorphism with respect to each complex
structure. This strong rigidity for aspherical hyper-Kähler manifolds is a
consequence of the Calabi-Yau theorem and the Cheeger-Gromoll splitting
theorem. For related result on complex hyperhermitian surfaces, see [11].

##### The paper is organized as follows

In Section 2, we prove Theorem 2 of the Introduction saying that $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)$ is finite whenever $\Gamma$ has no
normal solvable groups. In Section 3, we introduce a cohomology-invariant for
the group $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ (see Proposition 3.1)
to show the coincidence of the Lie groups $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ and
${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$ when $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ acts properly on $M$ in Theorem 3.3. In
particular, when $M$ is compact, $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$
is a compact Lie group. We prove Theorem 3 in Section 4. Section 5 concerns
the properties of standard $qc$-manifolds.

##  2\. ${{\mathsf{G}}}$-manifolds with compact automorphism groups

We study the structure of $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)$
for closed aspherical ${{\mathsf{G}}}$-manifolds.

##### Lifting Lemma

To prepare our proof we start our discussion with some well known general
setup. Let $\tilde{M}$ be the universal covering space of $M=\tilde{M}/\pi$
and denote $N_{{\rm Diff}(\tilde{M})}(\pi)$ the normalizer of $\pi$ in ${\rm
Diff}(\tilde{M})$. The conjugation

$\displaystyle\mu:N_{{\rm Diff}(\tilde{M})}(\pi){\rightarrow}{\rm Aut}(\pi)$

defined by $\mu(\tilde{f})(\gamma)=\tilde{f}\circ\gamma\circ\tilde{f}^{-1}$
$({}^{\forall}\,\gamma\in\pi)$ induces a homomorphism

$\varphi:{\rm Diff}(M)\to{\rm Out}(\pi)\;.$

######  Lemma 2.1 (see [27]).

There is an exact commutative diagram:

(2.1) $\begin{CD}111\\\ @V{}V{}V@V{}V{}V@V{}V{}V\\\ 1@>{}>{}>Z(\pi)@>{}>{}>\pi
@>{\mu}>{}>{\rm Inn}(\pi)\\\ @V{}V{}V@V{}V{}V@V{}V{}V\\\ 1@>{}>{}>Z_{{\rm
Diff}(\tilde{M})}(\pi)@>{}>{}>N_{{\rm Diff}(\tilde{M})}(\pi)@>{\mu}>{}>{\rm
Aut}(\pi)\\\ @V{\nu}V{}V@V{\nu}V{}V@V{}V{}V\\\ 1@>{}>{}>{\rm ker}\,\varphi
@>{}>{}>{\rm Diff}(M)@>{\varphi}>{}>{\rm Out}(\pi)\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ 111\\\ \end{CD}$

Here, $Z_{{\rm Diff}(\tilde{M})}(\pi)$ denotes the centralizer of $\pi$ in
${\rm Diff}(\tilde{M})$.

##### Application to automorphisms of aspherical manifolds

###### Proof of Theorem 2.

As we show now, replacing ${\rm Diff}(\tilde{M})$ in Lemma 2.1 by $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X)$, $N_{\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X)}(\Gamma)$ turns out to be discrete and the
homomorphism $\mu:N_{\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X)}(\Gamma){\rightarrow}{\rm Aut}(\Gamma)$
injective.

As in diagram (2.1), there is an exact sequence:

(2.2) $1{\rightarrow}\ker\varphi{\rightarrow}\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)\stackrel{{\scriptstyle\varphi}}{{{\longrightarrow}}}\mathop{\rm
Out}\nolimits(\Gamma)$

such that $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)^{0}\leq\ker\varphi$. Let
$Z_{\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(X)}(\Gamma)$ be the centralizer
of $\Gamma$ in $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(X)$. Then as can be
inferred from diagram (2.1), $\ker\varphi$ is associated with the covering
group extension:

$1{\rightarrow}\,Z(\Gamma){\rightarrow}\,Z_{\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X)}(\Gamma){\longrightarrow}\,\ker\varphi{\rightarrow}1.$

Since $Z(\Gamma)$ is the center of $\Gamma$, note $Z(\Gamma)=\\{1\\}$ by the
hypothesis so that

(2.3) $Z_{\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X)}(\Gamma)\cong\ker\varphi.$

Now $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)$ is compact by
Corollary 3.5. Since a compact connected Lie group acting on a closed
aspherical manifold is a torus (cf. Theorem B), $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)^{0}=T^{k}$, $k\geq 0$. Then the orbit
map ${\rm ev}(t)=tz$ of $T^{k}$ into $X/\Gamma$ at any point $z\in X/\Gamma$
induces an injective homomorphism $\displaystyle{\rm
ev}_{*}:{\mathbb{Z}}^{k}{\longrightarrow}\,\Gamma$ such that ${\rm
ev}_{*}({\mathbb{Z}}^{k})\leq Z(\Gamma)$ (cf. [16, Lemma 4.2]). As $Z(\Gamma)$
is trivial, by assumption, we have $k=0$. That is, $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)^{0}=\\{1\\}$. In particular
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)$ is a finite group.

Therefore, the subgroup $\ker\varphi$ is finite by (2.2), and so is
$Z_{\mathop{\rm Aut}\nolimits_{G}(X)}(\Gamma)$ by (2.3). As $Z_{\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X)}(\Gamma)$ is centralized by $\Gamma$, it
follows $\displaystyle Z_{\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X)}(\Gamma)=\\{1\\}$ by [8, Theorem 2]. Hence
$\ker\varphi=\\{1\\}$, that is $\displaystyle\varphi:\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma){\rightarrow}\mathop{\rm
Out}\nolimits(\Gamma)$ is injective.

Since the quotient of $N_{\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X)}(\Gamma)$ by $\Gamma$ is isomorphic to
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)$ from (2.1). As
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)$ is finite, $\Gamma$ is of
finite index in $N_{\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(X)}(\Gamma)$. ∎

Along the same direction as the above proof, we have:

######  Corollary 2.2.

Let $X/\Gamma$ be a closed aspherical manifold. Suppose that
$G\leq\operatorname*{Diff}\,(X/\Gamma)$ is a connected Lie group. If
$Z(\Gamma)$ is trivial (or finite), then $G$ is a simply connected solvable
Lie group.

##  3\. Invariant substructures for
${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$

Let $M$ be a parabolic ${\mathsf{G}}$-manifold. If $M$ is of $CR$\- or
$qc$-type, we shall prove that, when $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ acts properly, there exists a representative
form $\omega$ for the parabolic ${\mathsf{G}}$-structure such that
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ coincides with
${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M,\omega)$. (The obvious analogue also
holds for conformal structures defined by Riemannian metrics. See [6] and
references therein for both results).

##### Geometry associated with parabolic ${\mathsf{G}}$-structures

Whereas a conformal structure is equivalent with a conformal class of
Riemannian metrics, the classical geometries underlying the case (2) and (3)
parabolic geometries are considerably more involved. Let us thus briefly
recall the geometric data associated with case (2) and (3) parabolic
geometries:

In the case of $CR$-structures, we have a contact form $\omega$ on a connected
smooth manifold $M$, which is determined up to scaling with a positive
function, and a complex structure $J$ on the contact bundle $\ker\omega$ which
is compatible with $\omega$ in the sense that the Levi form $d\omega\circ J$
is a positive definite Hermitian form on $D$. These data define a _strictly
pseudo-convex_ $CR$-structure. Note that $\omega$ is defined up to a conformal
change with a positive function.

Let a $qc$-structure on a $4n+3$-manifold $M$ be given. This amounts to a
positive definite codimension three subbundle ${\mathsf{D}}$, which is non-
integrable and such that ${\mathsf{D}}+[{\mathsf{D}},{\mathsf{D}}]=TM$.
Moreover, there is a hypercomplex structure $\\{J_{k}\\}_{k=1}^{3}$ on
${\mathsf{D}}$, and an ${\rm Im}\,{\mathbb{H}}$-valued $1$-form
$\omega=\omega_{1}i+\omega_{2}j+\omega_{3}k$. It is also required that
${\mathsf{D}}=\mathop{\ker}\,\omega$ and the forms $d\omega_{k}\circ J_{k}$
are positive definite Hermitian forms. Note that $\omega$ is defined up to a
conformal change with a positive function and conjugation with $\mathop{\rm
Sp}\nolimits(1)$.

##### Description of the automorphism groups of parabolic
${\mathsf{G}}$-structures

The associated automorphism groups $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ are then described as follows (for cases (2)
and (3) compare [6, 23]):

(3.1) $\begin{cases}(1)\,\mathop{\rm
Conf}\nolimits(M)=\bigl{\\{}\,\alpha\in\operatorname*{Diff}\,(M)\mid\alpha^{*}g=u_{\alpha}\,g\,\bigr{\\}},\\\
(2)\,\mathop{\rm
Aut}\nolimits_{CR}(M,\\{\omega,J\\})=\\{\,\alpha\in\operatorname*{Diff}\,(M)\mid\alpha^{*}\omega=u_{\alpha}\,\omega,\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\alpha_{*}\circ J=J\circ\alpha_{*}|_{\ker\,\omega}\,\bigr{\\}},\\\
(3)\mathop{\rm
Aut}\nolimits_{qc}(M,(\omega,\\{J_{k}\\}_{k=1}^{3}))=\bigl{\\{}\,\alpha\in\operatorname*{Diff}\,(M)\mid\\\
\ \ \ \ \ \ \ \ \ \ \
\alpha^{*}\omega=u_{\alpha}\;a_{\alpha}\cdot\omega\cdot\overline{a_{\alpha}},\
\ \alpha_{*}\circ
J_{k}=\sum_{j=1}^{3}a_{kj}J_{j}\circ\alpha_{*}|_{\ker\,\omega}\,\bigr{\\}},\\\
\end{cases}$

where $u_{\alpha}\in C^{\infty}(M,{\mathbb{R}}^{+})$, $a_{\alpha}\in
C^{\infty}(M,\mathop{\rm Sp}\nolimits(1))$, and the matrix $(a_{kj})\in
C^{\infty}(M,\mathop{\rm SO}\nolimits(3))$ is given by the conjugation action
of $a_{\alpha}$ on ${\rm Im}\,{\mathbb{H}}$.

We would like to emphasize that the definition of $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ does not depend on the particular choice of
data $g$ or $\omega$ in their conformal class. In fact, the choice of $g$ or
$\omega$ amounts to choosing a representative geometry. The symmetries of the
representative geometry define a subgroup of $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$. These groups are the isometry group
$\mathop{\rm Iso}\nolimits(M,g)$, respectively the pseudo-Hermitian groups
${\operatorname*{Psh}\,}_{CR}(M,\\{\omega,J\\})$ and
${\operatorname*{Psh}\,}_{qc}(M,(\omega,\\{J_{k}\\}_{k=1}^{3}))$:

(3.2) $\begin{cases}(1)\,\mathop{\rm
Iso}\nolimits(M,g)=\bigl{\\{}\alpha\in\operatorname*{Diff}\,(M)\mid\alpha^{*}g=g\
\bigr{\\}},\\\
(2)\,{\operatorname*{Psh}\,}_{CR}(M,\omega)=\\{\alpha\in\operatorname*{Diff}\,(M)\mid\alpha^{*}\omega=\omega,\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\alpha_{*}\circ J=J\circ\alpha_{*}|_{\ker\,\omega}\bigr{\\}},\\\
(3)\,{\operatorname*{Psh}\,}_{qc}(M,\omega)=\bigl{\\{}\alpha\in\operatorname*{Diff}\,(M)\mid\alpha^{*}\omega=a_{\alpha}\cdot\omega\cdot\overline{a_{\alpha}},\\\
\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\alpha_{*}\circ
J_{k}=\sum_{j=1}^{3}a_{kj}J_{j}\circ\alpha_{*}|_{\ker\,\omega}\bigr{\\}}.\\\
\end{cases}$

Note that the groups in (3.2) vary considerably under a conformal change of
$g$, respectively $\omega$, while the group $\mathop{\rm
Aut}\nolimits_{\mathsf{G}}(M)$ is preserved.

###  3.1. Conformal invariant cohomology class

The space $C^{\infty}(M,{\mathbb{R}}^{+})$ of smooth positive functions on $M$
is endowed with an action of $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$,
where for $\alpha\in\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$, $f\in
C^{\infty}(M,{\mathbb{R}}^{+})$, we have

$(\alpha_{*}f)(x)=f(\alpha^{-1}x)\,\ (x\in M).$

Thus $C^{\infty}(M,{\mathbb{R}}^{+})$ is a smooth $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$-module. To any such module there is an
associated differentiable group cohomology $H^{*}_{d}$ for the Lie group
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ (see [6], for detailed
explanation). We explain now that the action of $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ on $C^{\infty}(M,{\mathbb{R}}^{+})$ gives
rise to a natural cohomology class that carries geometric information about
the dynamics of this action.

##### Construction of the associated cohomology class

######  Proposition 3.1.

For any closed subgroup $L$ of $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$
there is a natural cohomology class $[\lambda_{{\mathsf{G}}}]\in
H^{1}_{d}(L,C^{\infty}(M,{\mathbb{R}}^{+}))$, which is associated to the
parabolic ${\mathsf{G}}$-structure on $M$.

###### Proof.

Let $\alpha\in\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ such that
$\displaystyle\alpha^{*}g=u_{\alpha}g$ for a Riemannian metric,
$\alpha^{*}\omega=u_{\alpha}\omega$ for a contact form, or
$\displaystyle\alpha^{*}\omega=u_{\alpha}\,a_{\alpha}\cdot\omega\cdot\overline{a_{\alpha}}$
for a quaternionic contact form, representing the parabolic
${\mathsf{G}}$-structure on $M$. We construct $[\lambda_{\mathsf{G}}]$ for the
$qc$-group $\mathop{\rm Aut}\nolimits_{qc}(M)$ in place of $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$. (But the proof holds also for $\mathop{\rm
Conf}\nolimits(M)$, $\mathop{\rm Aut}\nolimits_{CR}(M)$, see [6])

Let $\alpha,\beta\in\mathop{\rm Aut}\nolimits_{qc}(M)$. We write
$\alpha\,\beta\in\mathop{\rm Aut}\nolimits_{qc}(M)$ for the composition of the
two $qc$-transformations. We calculate

$\begin{split}(\alpha\,\beta)^{*}\omega&=u_{\alpha\beta}\;a_{\alpha\beta}\cdot\omega\cdot\overline{a_{\alpha\beta}},\\\
\beta^{*}\alpha^{*}\omega&=\beta^{*}(u_{\alpha}\,a_{\alpha}\cdot\omega\cdot\overline{a_{\alpha}})=(\beta^{*}u_{\alpha}\,u_{\beta})(\beta^{*}a_{\alpha}\cdot
a_{\beta})\cdot\omega\cdot(\overline{\beta^{*}a_{\alpha}\cdot a_{\beta}}).\\\
\end{split}$

(Note that
$\displaystyle\beta^{*}\,\overline{a_{\alpha}}\,(x)=\overline{a_{\alpha}}\,(\beta
x)=\overline{a_{\alpha}(\beta x)}=\overline{\beta^{*}a_{\alpha}}\,(x)$).
Taking the norm, we have

$\displaystyle||(\alpha\beta)^{*}\omega||=u_{\alpha\beta}\,||\omega||=\beta^{*}u_{\alpha}\,u_{\beta}\,||\omega||\,.$

Thus the smooth maps $u_{\alpha},u_{\beta},u_{\alpha\beta}\in
C^{\infty}(M,{\mathbb{R}}^{+})$ satisfy

(3.3) $u_{\alpha\beta}=\beta^{*}u_{\alpha}\,u_{\beta}\ \ \mbox{on}\ M.$

Define $\lambda_{{\mathsf{G}}}=\lambda_{{\mathsf{G}},\omega}:\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)\to C^{\infty}(M,{\mathbb{R}}^{+})$ to be

(3.4) $\lambda_{{\mathsf{G}}}(\alpha)=\alpha_{*}u_{\alpha}.$

In particular,
$\displaystyle\lambda_{{\mathsf{G}}}(\alpha)(x)=u_{\alpha}(\alpha^{-1}x)$. We
observe that $\lambda_{{\mathsf{G}}}$ is a crossed homomorphism with respect
to the representation of $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ on
$C^{\infty}(M,{\mathbb{R}}^{+})$:

$\begin{split}\lambda_{{\mathsf{G}}}(\alpha\beta)\,(x)&=\,(\alpha\beta)_{*}u_{\alpha\beta}\,(x)=u_{\alpha\beta}\,(\beta^{-1}\alpha^{-1}x)\\\
&=\beta^{*}u_{\alpha}\,(\beta^{-1}\alpha^{-1}x)\;u_{\beta}\,(\beta^{-1}\alpha^{-1}x)\
\,(\text{by}\eqref{crossed})\\\
&=\lambda_{{\mathsf{G}}}(\alpha)\,(x)\;\alpha_{*}\lambda_{{\mathsf{G}}}(\beta)\,(x)=(\lambda_{{\mathsf{G}}}(\alpha)\;\alpha_{*}\lambda_{{\mathsf{G}}}\,(\beta))\,(x).\end{split}$

Hence,
$\displaystyle\lambda_{{\mathsf{G}}}(\alpha\beta)=\lambda_{{\mathsf{G}}}(\alpha)\cdot\alpha_{*}\lambda_{{\mathsf{G}}}(\beta)$,
that is, $\lambda_{{\mathsf{G}}}$ is a crossed homomorphism and thus a one-
cocycle for the differentiable cohomology of $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M)$ with coefficients in
$C^{\infty}(M,{\mathbb{R}}^{+})$. Let
$\displaystyle[\lambda_{{\mathsf{G}}}]\in H^{1}_{d}(\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M),\,C^{\infty}(M,{\mathbb{R}}^{+}))$ denote its
corresponding cohomology class.

We show $[\lambda_{{\mathsf{G}}}]$ is a conformal ${\mathsf{G}}$-invariant.
For $qc$-forms $\omega,\omega^{\prime}$, suppose that $\omega^{\prime}$ is
$qc$-equivalent to $\omega$, that is,

(3.5) $\omega^{\prime}=u\;b\cdot\omega\cdot\bar{b},\;\;\;\text{ for }u\in
C^{\infty}(M,{\mathbb{R}}^{+}),\;b\in C^{\infty}(M,\mathop{\rm
Sp}\nolimits(1))\;.$

For $\alpha\in\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$, write
$\displaystyle\alpha^{*}\omega^{\prime}=u^{\prime}_{\alpha}\;a^{\prime}_{\alpha}\cdot\omega^{\prime}\cdot\overline{a^{\prime}_{\alpha}}$.
Thus

$\lambda_{{\mathsf{G}},\omega^{\prime}}(\alpha)=\alpha_{*}u^{\prime}_{\alpha}\,.$

Then
$\displaystyle\alpha^{*}\omega^{\prime}=u^{\prime}_{\alpha}\;a^{\prime}_{\alpha}\cdot(u\;b\cdot\omega\cdot\bar{b})\cdot\overline{a^{\prime}_{\alpha}}=(u^{\prime}_{\alpha}u)\;(a^{\prime}_{\alpha}\cdot
b)\cdot\omega\cdot(\overline{a^{\prime}_{\alpha}\cdot b})$. Also

$\begin{split}\alpha^{*}\omega^{\prime}&=\alpha^{*}(u\;b\cdot\omega\cdot\bar{b})=(\alpha^{*}u\,u_{\alpha})\,(\alpha^{*}b\cdot
a_{\alpha})\cdot\omega\cdot(\overline{\alpha^{*}b\cdot
a_{\alpha}}).\end{split}$

Taking the norm $||\alpha^{*}\omega^{\prime}||$, it follows $\displaystyle
u^{\prime}_{\alpha}\,u=\alpha^{*}u\,u_{\alpha}$, that is,

$\displaystyle u^{\prime}_{\alpha}\,(\alpha^{*}u)^{-1}\,u=u_{\alpha}.$

This shows
$\displaystyle\alpha_{*}u^{\prime}_{\alpha}\cdot\delta^{0}(u)(\alpha)=\alpha_{*}u_{\alpha}$.
Hence,
$\displaystyle[\lambda_{{\mathsf{G}},\omega^{\prime}}]=[\lambda_{{\mathsf{G}},\omega}]$
and so the cohomology class $[\lambda_{{\mathsf{G}},\omega}]$ is a
quaternionic conformal invariant. ∎

Regarding the cohomology groups of $L$ with coefficients in
$C^{\infty}(M,{\mathbb{R}}^{+})$ we have the following important general fact:

######  Theorem 3.2 ([6, Theorem 10]).

Suppose that $L$ acts properly on $M$. Then

$H^{i}(L,C^{\infty}(M,{\mathbb{R}}^{+}))=\\{0\\},\;i\geq 1.$

Recall that ${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$ denotes the unique
maximal subgroup of $\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$ that acts
properly on $M$. We now prove:

######  Theorem 3.3.

Let $M$ be a parabolic ${\mathsf{G}}$-manifold of $CR$\- or $qc$-type. Then
there exists a representative form $\omega$ for the parabolic
${\mathsf{G}}$-structure such that

${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)={\operatorname*{Psh}\,}_{\mathsf{G}}(M,\omega)\;.$

###### Proof.

Put $L={\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$ for the following. Since
${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M)$ acts properly, Theorem 3.2 implies
$H^{1}(L,C^{\infty}(M,{\mathbb{R}}^{+}))=\\{0\\}$. Let $\eta$ be a
representative form for the parabolic ${\mathsf{G}}$-structure on $M$. Since
$H^{1}(L,C^{\infty}(M,{\mathbb{R}}^{+}))=\\{0\\}$, in particular,
$[\lambda_{{\mathsf{G}},\eta}]=0$, where $\lambda_{{\mathsf{G}},\eta}$ is a
one-cocycle for $L$.

Since $\alpha\in\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M)$, we can write
$\alpha^{*}\eta=u_{\alpha}\,a_{\alpha}\cdot\eta\cdot\overline{a_{\alpha}}$.
Thus the equation

$\lambda_{{\mathsf{G}},\eta}=\delta^{0}v,\text{ for some }v\in
C^{\infty}(M,{\mathbb{R}}^{+}),$

means that $\displaystyle\alpha_{*}u_{\alpha}=\alpha_{*}vv^{-1}$, $\alpha\in
L$. Or equivalently, $\displaystyle u_{\alpha}\cdot\alpha^{*}v=v$.

Put $\omega=v\,\eta$. Then it follows that

$\displaystyle\alpha^{*}\omega=\alpha^{*}v\;u_{\alpha}\,a_{\alpha}\cdot\eta\cdot\,\overline{a_{\alpha}}=v\;a_{\alpha}\cdot\eta\cdot\overline{a_{\alpha}}=a_{\alpha}\cdot\omega\cdot\overline{a_{\alpha}}.$

This shows that $\alpha\in{\operatorname*{Psh}\,}_{{\mathsf{G}}}(M,\omega)$.
That is, we have ${\operatorname*{Psh}\,}_{\mathsf{G}}(M)$ is contained in
${\operatorname*{Psh}\,}_{{\mathsf{G}}}(M,\omega)$. Since $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(M,\omega)$ is acting properly, by Lemma 3.4,
$\mathop{\rm Aut}\nolimits_{{\mathsf{G}}}(M,\omega)$ is contained in
${\operatorname*{Psh}\,}_{\mathsf{G}}(M)$. The theorem is proved. ∎

Next we note:

######  Lemma 3.4.

The subgroup ${\operatorname*{Psh}\,}_{qc}(M,\omega)$ of $\mathop{\rm
Aut}\nolimits_{\mathsf{G}}(M)$ preserves an associated Riemannian metric
$g_{\omega}$. In particular, ${\operatorname*{Psh}\,}_{qc}(M,\omega)$ acts
properly on $M$.

###### Proof.

$g_{\omega}(\boldsymbol{x},\boldsymbol{y})=\sum_{i=1}^{3}\omega_{i}(\boldsymbol{x})\cdot\omega_{i}(\boldsymbol{y})+d\omega_{1}(J_{1}\boldsymbol{x},\boldsymbol{y}),\,\text{
where }\boldsymbol{x},\boldsymbol{y}\in TM.\qed$

######  Corollary 3.5.

When $X/\Gamma$ is a closed aspherical ${\mathsf{G}}$-manifold, $\mathop{\rm
Aut}\nolimits_{{\mathsf{G}}}(X/\Gamma)$ is a compact Lie group. In particular,
$[\lambda_{{\mathsf{G}}}]=0$.

###### Proof.

By Theorem A, the automorphism group of any compact aspherical parabolic
${\mathsf{G}}$-manifold is acting properly. ∎

######  Remark 3.6.

Note that, as $\mathop{\rm
Aut}\nolimits_{CR}({\mathcal{N}})={\mathcal{N}}\rtimes{\rm
U}(n)\times{\mathbb{R}}^{+}$, is not acting properly on the Heisenberg group
${\mathcal{N}}$, $[\lambda_{CR}]\neq 0$ in $H^{1}_{d}(\mathop{\rm
Aut}\nolimits_{CR}({\mathcal{N}}),C^{\infty}({\mathcal{N}},{\mathbb{R}}^{+}))$.
On the other hand, if $X={\mathcal{N}}-\\{\boldsymbol{0}\\}$, then
$\mathop{\rm Aut}\nolimits_{CR}(X)={\rm U}(n)\times{\mathbb{R}}^{+}$ which
acts properly on $X$. Then $[\lambda_{CR}]\in H^{1}_{d}({\rm
U}(n)\times{\mathbb{R}}^{+},C^{\infty}(X,{\mathbb{R}}^{+}))=\\{0\\}$. The
quotient of $X$ by an infinite discrete subgroup of ${\rm
U}(n)\times{\mathbb{R}}^{+}$ is an infra-Hopf manifold.

##  4\. $CR$-manifolds $X/\pi$ with virtually solvable group $\pi$

###  4.1. Sasaki manifolds and Reeb flow

A Sasaki structure on $M$ is equivalent with a _standard_ $CR$-structure. For
any $CR$-structure $(\omega,J)$, the Reeb field $\xi$ is defined by the
conditions

$\omega(\xi)=1\text{ and }d\omega(\xi,\cdot)=0.$

A $CR$-structure is called standard if the flow of the Reeb field is contained
in the pseudo-Hermitian group $\operatorname*{Psh}\,_{CR}(M,\omega)$. Then the
metric

$g=\omega\cdot\omega+d\omega\circ J$

is called _Sasaki metric_. The group $\operatorname*{Psh}\,_{CR}(M,\omega)$ is
thus contained in the isometry group of the Sasaki metric $g$. Note further
that the Reeb flow is contained in the center of
$\operatorname*{Psh}\,_{CR}(M,\omega)$ (compare [7]).

##### The Reeb field generates $S^{1}$ on compact manifolds.

On a compact $CR$-manifold the Reeb flow always gives rise to a circle action.

######  Proposition 4.1.

Let $(M,\omega,J)$ be a closed strictly pseudoconvex standard $CR$-manifold.
Then the Reeb field generates an $S^{1}$-action on $M$.

###### Proof.

By the definition of standard $CR$-structure, the Reeb field $\xi$ generates a
one-parameter group ${{\mathsf{A}}}$ of $CR$-transformations for
$(\ker\,\omega,J)$. Let $\operatorname*{Psh}\,(M,(\omega,J))$ be the pseudo-
Hermitian group. Since $\operatorname*{Psh}\,(M,(\omega,J))\leq\mathop{\rm
Iso}\nolimits(M,g)$ for the Sasaki metric $g=\omega\cdot\omega+d\omega\circ
J$, $\operatorname*{Psh}\,(M,(\omega,J))$ is compact. If
$\displaystyle\bar{{\mathsf{A}}}$ is the closure of ${{\mathsf{A}}}$ in
$\operatorname*{Psh}\,(M,(\omega,J))$, then $\bar{A}$ is isomorphic to a
$k$-torus $T^{k}$. Let ${\mathcal{T}}^{k}$ be the distribution of vector
fields for $T^{k}$ on $M$. Consider the restriction
$\omega|_{{\mathcal{T}}^{k}}:{\mathcal{T}}^{k}{\rightarrow}{\mathbb{R}}$. Then
${\mathcal{T}}^{k}=\langle\xi\rangle\oplus\ker\,(\omega|_{{\mathcal{T}}^{k}})$.
(For this, recall $\omega(\xi)=1$, $d\omega(\xi,\cdot)=0$.) Let
$\boldsymbol{u}\in\ker\,(\omega|_{{\mathcal{T}}^{k}})$. Since $\ker\,\omega$
is $J$-invariant, $J\boldsymbol{u}\in\ker\,\omega$. As
$\boldsymbol{u}\in{\mathcal{T}}^{k}$, $\boldsymbol{u}$ generates a one-
parameter group of transformations $\\{\varphi_{t}\\}_{t\in{\mathbb{R}}}$
holomorphic on ${\mathsf{D}}$. Putting $p_{-t}=\varphi_{-t}p$, for a point
$p\in M$, note that
$\displaystyle[\boldsymbol{u},J\boldsymbol{u}]=\lim_{t{\rightarrow}0}({\varphi_{t}}_{*}J\boldsymbol{u}_{p_{-t}}-J\boldsymbol{u}_{p})/t=J\lim_{t{\rightarrow}0}({\varphi_{t}}_{*}\boldsymbol{u}_{p_{-t}}-\boldsymbol{u}_{p})/t=J[\boldsymbol{u},\boldsymbol{u}]=\boldsymbol{0}$.
Let $d\omega\circ J$ be the positive definite Levi form on $\ker\,\omega$,
then $\displaystyle
2d\omega(\boldsymbol{u},J\boldsymbol{u})=\boldsymbol{u}\omega(J\boldsymbol{u})-J\boldsymbol{u}\omega(\boldsymbol{u})-\omega([\boldsymbol{u},J\boldsymbol{u}])=0$.
Thus $\boldsymbol{u}=\boldsymbol{0}$. It follows ${{\mathsf{A}}}=S^{1}$. ∎

###  4.2. Aspherical Sasaki manifolds

Now let $M=X/\pi$ be a $2n+1$-dimensional closed aspherical manifold with a
standard $CR$-structure. Since $M$ is compact, the Reeb field $\xi$ generates
an $S^{1}$-action on $M$ (Proposition 4.1). Then it follows (cf. [7]) that

  1. (1)

A Sasaki structure $(\omega,J)$ induces a Kähler structure $(\Omega,J)$ on $W$
such that $d\omega=p^{*}\Omega$ and $J$ is an induced complex structure from
$(\ker\,\omega,J)$ with $p_{*}J=Jp_{*}$.

  2. (2)

The central group extension $\displaystyle
1\to{\mathbb{R}}\cap\pi\to\pi\stackrel{{\scriptstyle\phi}}{{{\longrightarrow}}}Q\to
1$ embeds into the pseudo-Hermitian group as in the diagram

(4.1)
$\setcounter{MaxMatrixCols}{11}\begin{CD}1@>{}>{}>{\mathbb{R}}@>{}>{}>\operatorname*{Psh}\,(X)@>{\phi}>{}>\mathop{\rm
Iso}\nolimits_{h}(W)@>{}>{}>1\\\ @A{}A{}A@A{}A{}A@A{}A{}A\\\
1@>{}>{}>{\mathbb{R}}\cap\pi @>{}>{}>\pi @>{\phi}>{}>Q@>{}>{}>1\end{CD}$

where the quotient group $Q=\pi\big{/}\,{\mathbb{R}}\cap\pi$ acts effectively
and properly discontinuously on $W$ as a group of Kähler isometries.

It follows from diagram (4.1) that $Q$ acts effectively and properly
discontinuously on $W$.

###  4.3. $S^{1}$-action on a closed aspherical manifold

Let $M=X/\pi$ be a closed aspherical manifold with an effective
$S^{1}$-action. Recall (cf. Section 2, proof of Theorem 2) that, for any $x\in
M$, the orbit map ${\rm ev}_{x}:S^{1}{\rightarrow}M$ defined by ${\rm
ev}_{x}(t)=t\cdot x$, induces an _injective_ homomorphism

${\rm ev}_{*}:\pi_{1}(S^{1},1)={\mathbb{Z}}\to\pi_{1}(M,x)=\pi,$

such that ${\rm ev}_{*}({\mathbb{Z}})\leq Z(\pi)$ is contained in the center
of $\pi$. This implies that the $S^{1}$-action lifts to a proper action of
${\mathbb{R}}$ on $X$, where $\pi$ commutes with ${\mathbb{R}}$. Since
$S^{1}={\mathbb{R}}/{\mathbb{Z}}$ and the action is effective, we thus have an
equivariant principal bundle on the universal cover $X$ of the form

(4.2)
$\begin{CD}({\mathbb{Z}}={\mathbb{R}}\cap\pi,{\mathbb{R}})@>{}>{}>(\pi,X)@>{}>{}>(Q=\pi\big{/}{\mathbb{R}}\cap\pi,W=X/{\mathbb{R}})\;.\end{CD}$

##### Associated principal bundle

We suppose now that $Q$ admits a torsion-free subgroup of finite index. Thus
we may choose a torsion-free normal subgroup $Q^{\prime}$ of finite index in
$Q$, such that $W/Q^{\prime}$ is a closed aspherical manifold. We put
$\pi^{\prime}=\phi^{-1}(Q^{\prime})$ for the preimage of $Q^{\prime}$ in
$\pi$. Then the central group extension

(4.3) $\begin{CD}1@>{}>{}>{\mathbb{Z}}={\mathbb{R}}\cap\pi
@>{}>{}>\pi^{\prime}@>{}>{}>Q^{\prime}@>{}>{}>1\end{CD}$

gives rise to a principal circle bundle

(4.4)
$\begin{CD}S^{1}={\mathbb{R}}/{\mathbb{Z}}@>{}>{}>P=X/\pi^{\prime}@>{p}>{}>B=W/Q^{\prime}\
.\end{CD}$

###  4.4. Standard $CR$-structures on circle bundles

As in Section 4.2, we now suppose that the contractible manifold $X$ has a
standard $CR$-structure $(\omega,J)$. We require further that the Reeb flow
generates the principal ${\mathbb{R}}$-action in (4.3). Also we assume that
the $CR$-structure is preserved by $\pi$, that is,
$\pi\leq\operatorname*{Psh}\,(X,\omega)$. And we impose that
$\pi\cap{\mathbb{R}}={\mathbb{Z}}$. This ensures that the principal action of
${\mathbb{R}}$ on $X$ descends to an _effective_ action of
$S^{1}={\mathbb{R}}\big{/}{\mathbb{Z}}$ on $X/\pi$, as in (4.4).

Since $\pi\leq\operatorname*{Psh}\,(X,\omega)$, the principal $S^{1}$-bundle
(4.4) with total space

$P=X/\pi^{\prime}$

inherits a compatible induced standard $CR$-structure $(\bar{\omega},J)$. The
Reeb field $\xi$ pushes down to the Reeb field $\bar{\xi}$ on $P$. Since the
action of $S^{1}={\mathbb{R}}/{\mathbb{Z}}$ is effective, $\bar{\xi}$ is also
the fundamental vector field of the principal $S^{1}$-action on $P$. This fact
implies that the induced contact form $\bar{\omega}$ is in fact a _connection
form_ for the principal circle bundle $P$ in (4.4).

Furthermore, $d\omega$ is the curvature form of the connection $\bar{\omega}$
and satisfies

$d\omega=p^{*}\bar{\Omega}\,,$

for a closed form $\bar{\Omega}$ on $B$. Since it arises as the curvature of
the connection form, it follows that the cohomology class of $\bar{\Omega}$ is
integral, and that

$e(B)=\,[\,\,\bar{\Omega}\,]\;\in H^{2}(B,{\mathbb{Z}})$

is the characteristic class of the bundle (4.4) (cf. [25] or [10, Section
2.2]).

Since we also have a $CR$-structure, $\bar{\Omega}$ is its associated Kähler
form on $B$.

##### Finite group action on $P$

Since the group $\pi$ is contained in $\operatorname*{Psh}\,(X,\omega)$, $\pi$
centralizes the ${\mathbb{R}}$-action on $X$, since it arises from the Reeb
flow. Therefore $\pi$ acts on $P$ by bundle automorphisms with respect to
(4.4). Furthermore, the action of $\pi$ on $P$ descends to the group

$\mu=Q/Q^{\prime}\,,$

which is acting on $B=W/Q^{\prime}$ by Kähler isometries. That is, $\mu$
preserves the Kähler form on $B$, and, in particular, it fixes the Kähler
class $e(B)$.

We further note:

######  Lemma 4.2.

The holomorphic action of the finite quotient group $\mu$ on $B$ is effective,
that is, the homomorphism $\displaystyle\mu\to{\rm hol}(B)$ is injective.

###### Proof.

By our construction, $Q$ is normalizing $Q^{\prime}$, and it has an effective
action on $W$. Since $W\to W/Q^{\prime}$ is a covering map, any element of $Q$
that acts trivially on $W/Q^{\prime}$ is a lift of the identity of
$W/Q^{\prime}$ with respect to this covering. Therefore, it must be in
$Q^{\prime}$, showing that $\displaystyle\mu\to{\rm hol}(B)$ is injective. ∎

###  4.5. Biholomorphism between $W$ and ${\mathbb{C}}^{n}$

By [5, Theorem 2.1], the aspherical Kähler manifold $W/Q^{\prime}$ is
biholomorphic to a complex euclidean space form
${\mathbb{C}}^{n}/\rho(Q^{\prime})$, where $\rho:Q^{\prime}\to
E_{{\mathbb{C}}}(n)={\mathbb{C}}^{n}\rtimes{\rm U}(n)$ is a faithful
representation. Note that $\rho(Q^{\prime})$ is a Bieberbach group. Therefore,
$\Lambda={\mathbb{C}}^{n}\cap\rho(Q^{\prime})$ is a maximal free abelian
normal subgroup of $\rho(Q^{\prime})$ and of finite index in
$\rho(Q^{\prime})$. Let

$\displaystyle H:\;W\to{\mathbb{C}}^{n}$

be a corresponding biholomorphism equivariant with respect to $\rho$. From
this, we see that (going down to a finite index subgroup if necessary) we may
choose $Q^{\prime}$ such that $\rho(Q^{\prime})=\Lambda$ is contained in
${\mathbb{C}}^{n}$. Then $X/Q^{\prime}$ is a complex torus biholomorphic to

$T^{n}_{\mathbb{C}}={\mathbb{C}}^{n}/\Lambda.$

Moreover, we have a covering action:

(4.5) $\begin{CD}\Lambda
@>{}>{}>(Q^{H},{\mathbb{C}}^{n})@>{}>{}>(Q^{H}/\Lambda,T^{n}_{\mathbb{C}})\end{CD}$

where $Q^{H}$ induces a holomorphic action of $Q^{H}/\Lambda$ on
$T^{n}_{\mathbb{C}}$. That is, there is a homomorphism
$\displaystyle\theta:Q^{H}/\Lambda\to{\rm hol}(T^{n}_{\mathbb{C}})$. Every
biholomorphic map of a complex torus $T^{n}_{\mathbb{C}}$ is induced by a
complex affine transformation of the vector space ${\mathbb{C}}^{n}$, see
[20]. Thus it follows that $Q^{H}$ is contained in $E_{{\mathbb{C}}}(n)$. In
particular, $\rho$ extends to a homomorphism $\rho:Q\to E_{{\mathbb{C}}}(n)$
inducing $\theta$.

Since $\rho(Q)$ is a crystallographic group, we may, in addition, arrange
things such that $\rho(Q^{\prime})$ equals $\Lambda$:

$\rho(Q^{\prime})=\Lambda=\rho(Q)\cap{\mathbb{C}}^{n}.$

Then, the finite group $Q^{H}/\Lambda$ maps injectively to ${\rm U}(n)$, so
that we have

(4.6) $\mu=Q^{H}/\Lambda\leq{\rm U}(n).$

##### Compatible choice of associated linear Hermitian form

Consider now the characteristic class $e(B)$ of the circle bundle (4.4). By
the biholomorphism $B\to T^{n}_{\mathbb{C}}$, $e(B)$ is transported to a class

$e(T^{n}_{\mathbb{C}})=e(B)^{H}\in H^{2}(T^{n}_{\mathbb{C}},{\mathbb{Z}}).$

Moreover, since $e(B)$ is a Kähler class, $e(T^{n}_{\mathbb{C}})$ is contained
in the Kähler cone of $T^{n}_{\mathbb{C}}$.

######  Lemma 4.3.

There exists a positive definite linear Hermitian two form
$\Omega_{{\mathbb{C}}^{n}}$ $($of type $(1,1))$ on ${\mathbb{C}}^{n}$ such
that its image $\bar{\Omega}_{\mathbb{C}}$ on $T^{n}_{\mathbb{C}}$ represents
the characteristic class $e(T^{n}_{\mathbb{C}})$. Then, we have,

(4.7) $e(T^{n}_{\mathbb{C}})=[\,\bar{\Omega}_{{\mathbb{C}}^{n}}\,]\in
H^{2}(T^{n}_{\mathbb{C}},{\mathbb{Z}})\,.$

###### Proof.

Let $g$ be a Kähler metric on $T^{n}_{\mathbb{C}}$ with Kähler form
$\bar{\Omega}$ such that the Kähler class
$e(T^{n}_{\mathbb{C}})=[\bar{\Omega}]$. Since $T^{n}_{\mathbb{C}}$ has trivial
canonical bundle its first Chern-class $c_{1}(T^{n}_{\mathbb{C}})$ vanishes.
Let $\Theta$ denote the Ricci form for $g$. Then
$0=c_{1}(T^{n}_{\mathbb{C}})=[\Theta]$. In particular, $\Theta$ is null-
cohomologous. In this situation the Calabi-Yau existence theorem for Kähler
metrics with prescibed Ricci curvature [9, Chapter 11] asserts that there
exists a unique Ricci flat Kähler metric $g^{\prime}$ on $T^{n}_{\mathbb{C}}$
with Kähler form $\bar{\Omega}^{\prime}$, satisfying
$[\bar{\Omega}^{\prime}]=[\bar{\Omega}]=e(T^{n}_{\mathbb{C}})$. As a
consequence of the Cheeger-Gromoll splitting and decomposition theorem for
Ricci-nonnegatively curved manifolds [14, 17] it follows that the only Ricci
flat Riemannian metrics on a torus (in fact on closed aspherical manifolds)
are flat metrics. This shows that $g^{\prime}$ is a flat Kähler metric on
$T^{n}_{\mathbb{C}}$ and thus invariant by the holomorphic action of
$T^{n}_{\mathbb{C}}$ on itself. Pulling back $\bar{\Omega}^{\prime}$ to a form
$\Omega^{\prime}$ on ${\mathbb{C}}^{n}$,
$\Omega_{{\mathbb{C}}^{n}}=\Omega^{\prime}$ is linear and it has the required
properties. ∎

Note that by this construction, $\rho(Q)$ preserves the positive definite
Hermitian form $\Omega_{{\mathbb{C}}^{n}}$, and we may thus put ${\rm
U}(n)={\rm U}(\Omega_{{\mathbb{C}}^{n}})$ on ${\mathbb{C}}^{n}$.

##### Equivalence of bundles over the complex torus

By the identification $W={\mathbb{C}}^{n}$ via the biholomorphism $H$ and our
construction, the bundle (4.2), now equates to an equivariant principal bundle

(4.8)
$\begin{CD}({\mathbb{Z}}={\mathbb{R}}\cap\pi^{\prime},{\mathbb{R}})@>{}>{}>(\pi^{\prime},X)@>{}>{}>(\Lambda=\pi^{\prime}\big{/}{\mathbb{R}}\cap\pi^{\prime},{\mathbb{C}}^{n}=X/{\mathbb{R}})\;\end{CD}$

that descends to a principal circle bundle

(4.9)
$\begin{CD}S^{1}={\mathbb{R}}/{\mathbb{Z}}@>{}>{}>P=X/\pi^{\prime}@>{p}>{}>T^{n}_{\mathbb{C}}={\mathbb{C}}^{n}/\Lambda\;,\end{CD}$

whose characteristic class is $e(T^{n}_{\mathbb{C}})$ as defined in (4.7).

###  4.6. Locally homogeneous standard $CR$-structure on $P$

For brevity, we put $\bar{\Omega}_{0}=\bar{\Omega}_{{\mathbb{C}}^{n}}$ for the
above linear Kähler form on $T^{n}_{\mathbb{C}}$. Since the bundle $P$ has
fundamental class $e(P)=[\bar{\Omega}_{0}]$, there exists a $CR$-structure
$(\bar{\eta},J)$ on $P$, with contact form $\bar{\eta}$, which is satisfying

(4.10) $d\bar{\eta}=p^{*}\,{\bar{\Omega}}_{0}.$

Indeed, let $\bar{\tau}\in\Omega^{1}(T^{n}_{\mathbb{C}})$ be any one-form such
that $\Omega_{0}-\Omega=d\tau$. Then

(4.11) $\bar{\eta}\,=\,\bar{\omega}+p^{*}\bar{\tau}$

satisfies (4.10). The forms $\bar{\omega}$ and $\bar{\eta}$ have the same Reeb
field, so that the Reeb field $\bar{\xi}$ generates the original
$S^{1}$-action on $P$. That is, $\bar{\eta}$ is a connection form for the
principal bundle (4.9).

The $CR$-structure $(\bar{\eta},J)$ on $P$ then pulls back to a
$\pi^{\prime}$-invariant $CR$-structure $(\eta,J)$ on $X$, with contact form
$\eta$, and Reeb field $\xi$, generating the ${\mathbb{R}}$-action on $X$.
Moreover, we have $d\eta=p^{*}\Omega_{0}$. (In particular, all the conditions
on $(X,\eta,J)$ in Section 4.4 are satisfied.) We note that the standard
$CR$-structure $(\eta,J)$ on $X$ is homogeneous. Indeed,
$\operatorname*{Psh}\,(X,\eta)$ acts transitively on $X$, since the
holomorphic isometries of
$(X,\eta,J)\big{/}\,{\mathbb{R}}=({\mathbb{C}}^{n},\Omega_{0})$ lift to
elements of $\operatorname*{Psh}\,(X,\eta)$ (see [7, Proposition 3.4]).

We show now that $X$ with its $CR$-structure is equivalent to the Heisenberg
group ${\mathcal{N}}$ equipped with its standard left-invariant
$CR$-structure. In fact, it follows that
$\operatorname*{Psh}\,(X,\eta)={\mathcal{N}}\rtimes{\rm U}(n)$, as is noted in
[7, Proposition 6.1 (2)]. Here, the Reeb flow ${\mathbb{R}}$ generates the
center of the Heisenberg group ${\mathcal{N}}$, and
${\mathcal{N}}/{\mathbb{R}}={\mathbb{C}}^{n}$. Note that, by construction,
$\pi^{\prime}$ is contained in $\operatorname*{Psh}\,(X,\eta,J)$. Therefore
$\pi^{\prime}$ is a discrete uniform subgroup of

$\operatorname*{Psh}\,(X,\eta,J)=\operatorname*{Psh}\,({\mathcal{N}})={\mathcal{N}}\rtimes{\rm
U}(n).$

Note further that $\pi^{\prime}$ is a nilpotent group, since it is a central
extension of $\Lambda$. Therefore $\pi^{\prime}\leq{\mathcal{N}}$ is a uniform
lattice in ${\mathcal{N}}$, by the Bieberbach theorem for nilmanifolds [1,
24]. In particular, the orbit map ${\mathcal{N}}\to X$ gives an
$S^{1}$-equivariant diffeomorphism

${\mathcal{N}}/\pi^{\prime}\to X/\pi^{\prime}$

over the base space $T^{n}_{\mathbb{C}}$. This shows that $X/\pi^{\prime}$ is
a Heisenberg nilmanifold.

###  4.7. Locally homogeneous standard $CR$-structure on $X/\pi$

As we have seen, the group $\mu=\pi/\pi^{\prime}$ acts on the total space $P$
of the principal circle bundle (4.9) by bundle automorphisms, and the induced
action of $\mu$ on $T^{n}_{\mathbb{C}}$ fixes the curvature form
$\bar{\Omega}_{0}$.

For the final step in the proof of Theorem 3, we show now that there exists a
connection form with curvature $\bar{\Omega}$ that is fixed by the group
$\mu$:

######  Proposition 4.4.

There exists a connection form $\bar{\vartheta}$ for the principal circle
bundle (4.9), such that

  1. (1)

$d\bar{\vartheta}=p^{*}\,{\bar{\Omega}}_{0}$.

  2. (2)

$\mu$ is contained in $\operatorname*{Psh}\,(P,\bar{\vartheta},J)$.

###### Proof.

Let ${\mathcal{A}}(\bar{\Omega}_{0})=\\{\bar{\eta}\in\Omega^{1}(P)\mid
d\bar{\eta}=p^{*}\,{\bar{\Omega}}_{0}\\}$ be the space of connection forms for
the bundle (4.11) with curvature $\bar{\Omega}_{0}$. Choose a base point
$\bar{\eta}_{0}\in{\mathcal{A}}(\bar{\Omega}_{0})$. Then
${\mathcal{A}}(\bar{\Omega}_{0})=\bar{\eta}_{0}+\\{p^{*}\tau\mid\tau\in
Z^{1}(T^{n}_{\mathbb{C}})\\}$ is an affine space over the vector space of
closed one-forms $Z^{1}(T^{n}_{\mathbb{C}})$. Since the curvature
$\bar{\Omega}_{0}$ is preserved by $\mu$, for any $g\in\mu$,
$\bar{\eta}\in{\mathcal{A}}(\bar{\Omega}_{0})$, it follows
$g^{*}\bar{\eta}\in{\mathcal{A}}(\bar{\Omega}_{0})$. Therefore the group $\mu$
acts on ${\mathcal{A}}(\bar{\Omega}_{0})$. For $g\in\mu$, define

$c(g)=g^{*}\bar{\eta}_{0}-\bar{\eta}_{0}\,\in Z^{1}(T^{n}_{\mathbb{C}})\;.$

Since
$c(g_{1}g_{2})=g_{2}^{*}g_{1}^{*}\bar{\eta}_{0}-\bar{\eta}_{0}=g_{2}^{*}c(g_{1})+c(g_{2})$,
$c:\mu\to Z^{1}(T^{n}_{\mathbb{C}})$ defines a one-cocycle for the natural
representation of $\mu$ on $Z^{1}(T^{n}_{\mathbb{C}})$. The action of $\mu$ on
${\mathcal{A}}(\bar{\Omega}_{0})$ is by affine transformations, since, for
$\bar{\eta}\in{\mathcal{A}}(\bar{\Omega}_{0})$, we have

$g^{*}\bar{\eta}=\bar{\eta}_{0}+\,g^{*}(\bar{\eta}-\bar{\eta}_{0})+c(g).$

There is a corresponding cohomology class $[c]\in
H^{1}(\mu,Z^{1}(T^{n}_{\mathbb{C}}))$ associated to the one-cocycle $c$. Since
$\mu$ is finite, we have $H^{1}(\mu,Z^{1}(T^{n}_{\mathbb{C}}))=\\{0\\}$, so
$c$ must be a coboundary. This implies that there exists $\tau\in
Z^{1}(T^{n}_{\mathbb{C}})$ such that $c(g)=g^{*}\tau-\tau$. As a consequence,
$\bar{\vartheta}=\bar{\eta}_{0}-\tau$ is a fixed point for the action of
$\mu$.∎

Pulling back the connection form $\bar{\vartheta}$ to a connection form
$\vartheta$ on the covering bundle (4.8), we note:

######  Corollary 4.5.

There exists on $X$ a connection form $\vartheta$ for the bundle (4.2), with
$d\vartheta=\Omega_{0}$, such that the corresponding $CR$-structure on $X$
satisfies

$\pi\leq\operatorname*{Psh}\,(X,\vartheta,J).$

As above, the $CR$-structure $(X,\vartheta,J)$ is homogeneous and
$\operatorname*{Psh}\,(X,\vartheta,J)=\operatorname*{Psh}\,({\mathcal{N}})={\mathcal{N}}\rtimes{\rm
U}(n)$. Therefore $X/\pi$ is an infra-nilmanifold that is finitely covered by
the nilmanifold $X/\pi^{\prime}$. This concludes the proof of Theorem 3.

##  5\. Indication to the proof of Theorem 4

###  5.1. Quaternionic contact structures

Let $X$ be a $4n+3$-dimensional smooth manifold. A _quaternionic contact
structure_ is a codimension $3$-subbundle ${\mathsf{D}}$ on $X$ which
satisfies that
$\displaystyle{\mathsf{D}}\oplus[{\mathsf{D}},{\mathsf{D}}]=TX$. In addition
the following conditions are required:

There exists a non-degenerate ${\rm Im}\,{\mathbb{H}}$-valued $1$-form
$\omega=\omega_{1}i+\omega_{2}j+\omega_{3}k$ called a _quaternionic contact
form_ on $X$ such that

  1. (1)

$\displaystyle\ker\,\omega=\mathop{\cap}_{i=1}^{3}\ker\,\omega_{i}={\mathsf{D}}$.

  2. (2)

$\displaystyle\omega\mathop{\wedge}\omega\mathop{\wedge}\omega\,\wedge\overbrace{d\omega\wedge\cdots\wedge
d\omega}^{n}\neq 0$ on $X$.

The non-degeneracy of $d\omega_{k}$, $k=1,2,3$, on ${\mathsf{D}}$ defines the
bundle of endomorphisms $\\{J_{1},J_{2},J_{3}\\}$:

(5.1)
$J_{k}=(d\omega_{j}|_{{\mathsf{D}}})^{-1}\circ(d\omega_{i}|_{{\mathsf{D}}}):{{\mathsf{D}}}{\rightarrow}\,{{\mathsf{D}}},\
\,(i,j,k)\sim(1,2,3)$

which constitutes a _hypercomplex structure_ on ${\mathsf{D}}$. Note that the
Levi form

$d\omega_{i}\circ
J_{i}:{\mathsf{D}}\times{\mathsf{D}}{\longrightarrow}\,{\mathbb{R}},\;i=1,2,3$

is a positive definite symmetric bilinear form on ${\mathsf{D}}$. Then
$(X,{\mathsf{D}},\omega,\\{J_{i}\\}_{i=1}^{3})$ is said to be a positive
definite $qc$-manifold.

##### Standard $qc$-manifolds and three-Sasaki manifolds

For any $qc$-manifold $M$, let ${\mathcal{T}}=\\{\xi_{1},\xi_{2},\xi_{3}\\}$
denote the three-dimensional integrable distribution complementary to the
codimension three subbundle ${\mathsf{D}}$ on $M$ determined by the
$qc$-structure, called _$qck$ -distribution_ (see Section 3). If
${\mathcal{T}}$ generates a subgroup of ${\operatorname*{Psh}\,}_{qc}(M)$,
then $M$ is called a _standard_ $qc$-manifold.

######  Proposition 5.1.

Let $(M,\omega,\\{J_{i}\\}_{i=1}^{3})$ be a closed strictly pseudoconvex
standard $qc$-manifold. Then the Reeb fields generate a compact Lie group
action of $\mathop{\rm SU}\nolimits(2)$ or the torus group $T^{3}$ on $M$.

See [23, Proposition 4.5] for the proof. If the Reeb fields generate an
$\mathop{\rm SU}\nolimits(2)$-action then $M$ is called a _three-Sasaki
manifold_ (sometimes also a quaternionic $CR$-manifold [3]).

###  5.2. Aspherical standard $qc$-manifolds

Let $X/\pi$ be a $4n+3$-dimensional positive definite closed aspherical
_standard_ $qc$-manifold. Since $X/\pi$ is aspherical, the $qck$-distribution
$\hat{\mathcal{T}}=\\{\hat{\xi}_{1},\hat{\xi}_{2},\hat{\xi}_{3}\\}$ on $X/\pi$
generates a compact Lie group action by Proposition 5.1. Since $X/\pi$ is
aspherical, the flow is given by a three-torus
$T^{3}\leq{\operatorname*{Psh}\,}_{qc}(X/\pi)$. Moreover, the $T^{3}$-action
lifts to a proper ${\mathbb{R}}^{3}$-action on $X$ (as in Section 4.3 in the
case of circle actions).

This gives rise to an equivariant principal bundle over
$W=X/{\mathbb{R}}^{3}$:

$\displaystyle({\mathbb{R}}^{3}\cap\pi,{\mathbb{R}}^{3})\to\,(\pi,X)\stackrel{{\scriptstyle
p}}{{{\longrightarrow}}}\,(Q,W)\;.$

Then it follows (cf. [23]) that

  1. (1)

The standard qc-structure $(\omega,\\{J_{i}\\}_{i=1}^{3})$ induces a hyper-
Kähler structure $(\Omega,J=\\{J_{i}\\}_{i=1}^{3})$ on $W$, such that
$d\omega=p^{*}\Omega$, where $\Omega=\Omega_{1}i+\Omega_{2}j+\Omega_{3}k$ and
$J$ is an induced hypercomplex structure from $(\ker\,\omega,J)$ with
$p_{*}J=Jp_{*}$. Here $\Omega_{i}$ is a Kähler form with respect to $J_{i}$,
$i=1,2,3$.

  2. (2)

The central group extension $\displaystyle
1\to{\mathbb{R}}^{3}\cap\pi\to\pi\stackrel{{\scriptstyle\phi}}{{{\longrightarrow}}}Q\to
1$ embeds into the pseudo-Hermitian group of the $qc$-structure as in the
diagram

(5.2)
$\setcounter{MaxMatrixCols}{11}\begin{CD}1@>{}>{}>{\mathbb{R}}^{3}@>{}>{}>{\operatorname*{Psh}\,}_{qc}(X)@>{\phi}>{}>{\mathop{\rm
Iso}\nolimits}_{hK}(W)@>{}>{}>1\\\ @A{}A{}A@A{}A{}A@A{}A{}A\\\
1@>{}>{}>{\mathbb{R}}^{3}\cap\pi @>{}>{}>\pi @>{\phi}>{}>Q@>{}>{}>1\end{CD}$

where the quotient group $Q=\pi\big{/}\,{\mathbb{R}}^{3}\cap\pi$ acts
effectively and properly discontinuously on $W$, and as a group of hyper-
Kähler isometries for $(\Omega,J)$.

  3. (3)

$X$ is $qc$-homogeneous if and only if $W$ is hyper-Kähler homogeneous ([23,
Proposition C]).

##### Detailed explanation of (5.2)

Let $\omega=\omega_{1}i+\omega_{2}j+\omega_{3}k$ be a lift of the $qc$-form of
$X/\pi$ to the universal covering (cf. Section 3). Then $\omega$ is a
$\pi$-invariant non-degenerate ${\rm Im}\,{\mathbb{H}}$-valued $1$-form such
that $\pi\leq\operatorname*{Psh}\,_{qc}(X,(\omega,\\{J_{i}\\}_{j=1}^{3})$ as
in (3) of (3.2). Let ${\mathcal{T}}=\\{\xi_{1},\xi_{2},\xi_{3}\\}$ be a lift
of $\hat{\mathcal{T}}$ which generates a proper ${\mathbb{R}}^{3}$-action on
$X$. Since $\pi$ centralizes ${\mathbb{R}}^{3}$, for every $\gamma\in\pi$, it
follows $\gamma_{*}\xi_{i}=\xi_{i}$ $(i=1,2,3)$. Noting the action of $\pi$ on
$\omega$ from (3) of (3.2), each element $\gamma\in\pi$ satisfies

(5.3) $\gamma^{*}\omega=\omega,\ \gamma_{*}J_{i}=J_{i}\gamma_{*}.$

Recall that the $qc$-structure of
$\displaystyle(X,{\mathbb{R}}^{3},\\{\omega_{i},J_{i}\\}_{i=1}^{3},g_{\omega})$
induces a simply connected complete hyper-Kähler manifold
$\displaystyle(W,\\{\Omega_{i},J_{i}\\}_{i=1}^{3},g)$ for which
$p^{*}\Omega=d\omega$ where $\omega=\omega_{1}i+\omega_{2}j+\omega_{3}k$ is
the $\pi$-invariant one-form on $X$ and $\Omega$ is a $Q$-invariant two-form
on $W$:

(5.4) $\Omega=\Omega_{1}i+\Omega_{2}j+\Omega_{3}k.$

Furthermore,
$g_{\omega}=\sum_{i=1}^{3}\omega_{i}\cdot\omega_{i}+d\omega_{1}\circ J_{1}$ is
a canonical Riemannian metric on $X$ and $g=\Omega_{i}\circ J_{i}$ $(i=1,2,3)$
is a hyper-Kähler metric on $W$. By the equation $p^{*}\Omega=d\omega$ and
(5.3), each $\alpha\in Q$ preserves the hyper-Kähler struture on $W$:

(5.5) $\alpha^{*}\Omega=\Omega,\ \,\alpha_{*}J_{i}=J_{i}\alpha_{*}.$

In paricular, $Q$ acts as Kähler isometries of $(W,(\Omega_{i},J_{i}))$.

####  5.2.1. Associated principal three-torus bundle

We suppose now that $Q$ admits a torsion-free subgroup of finite index. Thus
we may choose a torsion-free normal subgroup $Q^{\prime}$ of finite index in
$Q$, such that $W/Q^{\prime}$ is a closed aspherical manifold. We put
$\pi^{\prime}=\phi^{-1}(Q^{\prime})$ for the preimage of $Q^{\prime}$ in
$\pi$. Then the central group extension

(5.6) $\begin{CD}1@>{}>{}>{\mathbb{Z}}^{3}={\mathbb{R}}^{3}\cap\pi
@>{}>{}>\pi^{\prime}@>{}>{}>Q^{\prime}@>{}>{}>1\end{CD}$

gives rise to a principal torus bundle

(5.7)
$\begin{CD}T^{3}={\mathbb{R}}^{3}/{\mathbb{Z}}^{3}@>{}>{}>P=X/\pi^{\prime}@>{p}>{}>B=W/Q^{\prime}\
.\end{CD}$

###  5.3. Proof of Lemma D

Let $g$ be hyper-Kähler with hyper-Kähler form $\Omega$ (compare (5.4)) on
$M$. Since $g$ is hyper-Kähler its holonomy group is contained in $\mathop{\rm
Sp}\nolimits(n)$. In particular, it is contained in $\mathrm{SU}(n)$, which
implies that $g$ is a Ricci- flat Kähler metric on $M$ (e.g. [9, Proposition
10.29]). In view of the fact that $M$ is aspherical all Ricci flat metrics are
flat [17]. We therefore have established that $g$ is a flat hyper-Kähler
metric. In particular, its universal covering space must be isometric to
${\mathbb{H}}^{n}$ with linear Kähler structure $\Omega$.

###  5.4. Hypercomplex isometric isomorphim

Applying Lemma D in the introduction to the hyper-Kähler manifold
$B=W/Q^{\prime}$ asserts that $B=W/Q^{\prime}$ with the hyper-Kähler structure
(5.4) is hyper-holomorphically isometric to a flat hyper-Kähler torus
$T^{n}_{\mathbb{H}}={\mathbb{H}}^{n}/\Lambda$, where
$\Lambda\leq{\mathbb{H}}^{n}$ is a lattice.

###  5.5. Conclusion of the proof

Using basic methods developed in [23] the remaining part of the proof of
Theorem 4 follows a similar but simplified procedure as that of Section 4. The
key step is to establish that the universal covering $X$ with its
$qc$-structure is homogeneous and arises from a quaternionic Heisenberg group,
that is $X$ is $qc$-isometric to a $qc$-Heisenberg group with its standard
$qc$-structure. The details shall be presented elsewhere.

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