Anti-regular functions on hyperkähler and quaternionic Kähler manifolds are
studied in [CL1, CL2, LZ], under the name quaternionic maps, in connection
with minimal surfaces and maps between quaternionic Kähler manifolds
preserving the sphere of almost complex structures. Thus, the anti-regular
functions considered here are quaternionic maps between $\mathbb{H}^{n}$ and
$\mathbb{H}$ with a suitable choice of the coordinates.

######  Definition 6.7.

A real-differentiable function $f:\mathbb{H}^{n}\mapsto\mathbb{R}$ is called
_quaternionic pluriharmonic_ (_Q-pluriharmonic_ for short) if it is the real
part of an anti-regular function.

The anti-regular functions and their real part play a significant role in the
theory of hypercomplex manifold as well as in the theory of quaternionic
contact (hypercomplex contact) manifolds as we shall see further in the paper.
We need a real expression of the second order differential operator
$\mathcal{D}_{\alpha}\ \overline{\mathcal{D}}_{\beta}\ f$ acting on a real
function $f$.

We use the standard hypercomplex structure on $\mathbb{H}^{n}$ determined by
the action of the imaginary quaternions

$\displaystyle I_{1}\,\,dt^{\alpha}\ =\ \,dx^{\alpha},$ $\displaystyle\quad
I_{1}\,\,dy^{\alpha}\ =\ \,dz^{\alpha},\quad\quad I_{2}\,dt^{\alpha}\ =\
\,dy^{\alpha},\quad I_{2}\,\,dx^{\alpha}\ =\ -\,dz^{\alpha}$ $\displaystyle
I_{3}\,dt^{\alpha}\ =\ \,dz^{\alpha},\quad\quad I_{3}\,\,dx^{\alpha}\ =\
\,dy^{\alpha}.$

We recall a convention. For any p-form $\psi$ we consider the p-form
$I_{s}\psi$ and three (p+1)-forms $d_{s}\psi,\quad s=1,2,3$ defined by
$I_{s}\psi(X_{1},\dots,X_{p}):=(-1)^{p}\psi(I_{s}X_{1},\dots,I_{s}X_{p}),\qquad
d_{s}\psi:=(-1)^{p}I_{s}dI_{s}\psi.$ Consider the second order differential
operators $DD_{I_{s}}$ acting on the exterior algebra defined by [HP]

(6.6) $DD_{I_{i}}:=dd_{i}+d_{j}d_{k}=dd_{i}-I_{j}dd_{i}=dd_{i}-I_{k}dd_{i}.$

######  Proposition 6.8.

Let $f$ be a real-differentiable function
$f:\mathbb{H}^{n}\rightarrow\mathbb{R}$. The following conditions are
equivalent

  1. i)

$f$ is Q-pluriharmonic, i.e. it is the real part of an anti-regular function;

  2. ii)

$DD_{I_{s}}\,f\ =\ 0,\quad s=1,2,3.$;

  3. iii)

$\mathcal{D}_{\alpha}\ \overline{\mathcal{D}}_{\beta}\ f\ =\ 0$ for every
$\alpha,\beta\in\\{1,\dots,n\\}$, where $\mathcal{D}_{\alpha}\ \ =\
{\partial_{t_{\alpha}}}{}\ -\ i{\partial_{x_{\alpha}}}{}\ -\
j{\partial_{y_{\alpha}}}{}\ -\ k{\partial_{z_{\alpha}}}{}$,

  4. iv)

$f$ satisfies the following system of PDEs

$\displaystyle{f}_{t_{\beta}{t_{\alpha}}}+{f}_{x_{\beta}{x_{\alpha}}}+{f}_{y_{\beta}{y_{\alpha}}}+{f}_{z_{\beta}{z_{\alpha}}}\
=\
0,\quad-{f}_{x_{\beta}{t_{\alpha}}}+{f}_{t_{\beta}{x_{\alpha}}}-{f}_{y_{\beta}{z_{\alpha}}}+{f}_{z_{\beta}{y_{\alpha}}}\
=\ 0,$
$\displaystyle{f}_{t_{\beta}{y_{\alpha}}}+{f}_{x_{\beta}{z_{\alpha}}}-{f}_{y_{\beta}{t_{\alpha}}}-{f}_{z_{\beta}{x_{\alpha}}}\
=\
0,\quad{f}_{t_{\beta}{z_{\alpha}}}-{f}_{x_{\beta}{y_{\alpha}}}+{f}_{y_{\beta}{x_{\alpha}}}-{f}_{z_{\beta}{t_{\alpha}}}\
=\ 0.$

###### Proof.

A simple calculation of $\mathcal{D}_{\beta}\ \overline{\mathcal{D}}_{\alpha}\
f$ gives the equivalence between iii) and iv).

Next, we shall show that ii) is equivalent to iii). As
$df={\partial_{t_{\alpha}}}{f}\,dt^{\alpha}+{\partial_{x_{\alpha}}}{f}\,dx^{\alpha}+{\partial_{y_{\alpha}}}{f}\,dy^{\alpha}+{\partial_{z_{\alpha}}}{f}\,dz^{\alpha}$
we have
$I_{1}df={\partial_{t_{\alpha}}}{f}\,dx^{\alpha}-{\partial_{x_{\alpha}}}{f}\,dt^{\alpha}+{\partial_{y_{\alpha}}}{f}\,dz^{\alpha}-{\partial_{z_{\alpha}}}{f}\,dy^{\alpha}.$
A routine calculation gives the following formula

(6.7) $D\,D_{I_{1}}\,f\ =\ \sum_{\alpha,\beta}\Re(\mathcal{D}_{\beta}\
\overline{\mathcal{D}}_{\alpha}\
f)\,[\,dt^{\alpha}\wedge\,dx^{\beta}+\,dy^{\alpha}\wedge\,dz^{\beta}]\\\
+\sum_{\alpha<\beta}\Re(i\mathcal{D}_{\beta}\ \overline{\mathcal{D}}_{\alpha}\
f)\,[-\,dt^{\alpha}\wedge\,dt^{\beta}-\,dx^{\alpha}\wedge\,dx^{\beta}+\,dy^{\alpha}\wedge\,dy^{\beta}+\,dz^{\alpha}\wedge\,dz^{\beta}]\\\
+\sum_{\alpha,\beta}\Re(j\mathcal{D}_{\beta}\ \overline{\mathcal{D}}_{\alpha}\
f)\,[\,dt^{\alpha}\wedge\,dz^{\beta}-\,dx^{\alpha}\wedge\,dy^{\beta}]\ -\
\sum_{\alpha,\beta}\Re(k\mathcal{D}_{\beta}\ \overline{\mathcal{D}}_{\alpha}\
f)\,[\,dt^{\alpha}\wedge\,dy^{\beta}+\,dx^{\alpha}\wedge\,dz^{\beta}].$

Similar formulas hold for $DD_{I_{2}}$ and $DD_{I_{3}}$. Hence, the
equivalence of ii) and iii) follows.

The proof of the implication iii) implies i) is analogous to the proof of the
corresponding implication in Proposition 6.5. Define $F(q)\ =\ f(q)\ +\
\Im\,\int_{o}^{1}s^{2}\,(\overline{\mathcal{D}}_{\beta}\
f)(sq)\,q^{\overline{\beta}}\,ds,$ and a small calculation shows that this
defines an anti-regular function, i.e., $\mathcal{D}_{\alpha}\ F\ =\ 0$ for
every $\alpha$.

In order to see that iii) follows from i) we can proceed as in Proposition 6.5
and hence we skip the details. See also another proof in Proposition 6.11 ∎

######  Remark 6.9.

We note that Proposition 6.5 and Proposition 6.8 imply that the real part of a
regular function is not in the kernel of the operators $DD_{I_{s}}$ which is
one of the main difference between regular and anti-regular function.

###  6.2. Quaternionic pluriharmonic functions on hypercomplex manifold

We recall that a hypercomplex manifold is a smooth $4n$-dimensional manifold
$M$ together with a triple $(I_{1},I_{2},I_{3})$ of integrable almost complex
structures satisfying the quaternionic relations
$I_{1}I_{2}=-I_{2}I_{1}=I_{3}$. The second order differential operators
$DD_{I_{i}}$ defined in [HP] by (6.21) having the origin in the papers [Sal1,
Sal2, CSal] play an important róle in the theory of quaternionic
plurisubharmonic functions (i.e. a real function for which
$DD_{I_{s}}(.,I_{s}.)$ is positive definite) on hypercomplex manifold [A1, A2,
V, AV, A3] as well as the potential theory of HKT-manifolds. We recall that
Riemannian metric $g$ on a hypercomplex manifold compatible with the three
complex structures is said to be HKT-metric [HP] if the three corresponding
Kähler forms $\Omega_{s}=g(I_{s}.,.)$ satisfy
$d_{1}\Omega_{1}=d_{2}\Omega_{2}=d_{3}\Omega_{3}$. A smooth real function is a
HKT-potential if locally it generates the three Kähler forms,
$\Omega_{s}=DD_{I_{s}}f$ [MS, GP], in particular such a function is
quaternionic plurisubharmonic. The existence of a HKT potential on any HKT
metric on $\mathbb{H}^{n}$ is proved in [MS] and for any HKT metric in [BS].

Regular functions on hypercomplex manifold are studied from analytical point
[Per1, Per2], from algebraic point [Joy, Q]. However, as we have already
mentioned, regular functions are not the appropriate functions for our
purposes mainly because they have no direct connection with the second order
differential operator $DD_{I_{s}}$.

Here we consider anti-regular functions and their real parts on hypercomplex
manifold.

######  Definition 6.10.

Let $(M,I_{1},I_{2},I_{3})$ be a hypercomplex manifold. A quaternionic valued
function $F:M\longrightarrow f+iw+ju+kv\in\mathbb{H}$ is said to be _anti-
regular_ if any one of the following relations between the differentials of
the coordinates hold

(6.8) $\displaystyle df=d_{1}w+d_{2}u+d_{3}v\qquad d_{1}f=-dw+d_{3}u-d_{2}v$
$\displaystyle d_{2}f=-d_{3}w-du+d_{1}v\qquad d_{3}f=d_{2}w-d_{1}u-dv.$

A real valued function $f:M\longrightarrow\mathbb{R}$ is said to be
_quaternionic pluriharmonic_ ( or _Q-pluriharmonic_) if it is the real part of
an anti-regular function.

Observe that the system (6.5) is equivalent to (6.8). We have the hypercomplex
manifold analogue of Proposition 6.8

######  Proposition 6.11.

Let $(M,I_{1},I_{2},I_{3})$ be a hypercomplex manifold and let $f$ be a real-
differentiable function on $M$, $f:M\longrightarrow\mathbb{R}$. The following
conditions are equivalent

  1. i)

$f$ is Q-pluriharmonic, i.e. it is the real part of an anti-regular function;

  2. ii)

$DD_{I_{s}}\,f\ =\ 0,\quad s=1,2,3.$;

###### Proof.

It is easy to verify that if each $I_{s}$ is integrable almost complex
structure then we have the identities [HP]

(6.9) $dd_{s}+d_{s}d=0,\qquad d_{s}d_{r}+d_{r}d_{s}=0,s,r=1,2,3.$

Using the commutation relations (6.9), we get readily that i) implies ii). For
example, (6.8) yields

$\displaystyle
dd_{1}f+d_{2}d_{3}f+d^{2}w-d_{2}^{2}w-dd_{3}u+d_{2}d_{1}u+dd_{2}v+d_{2}dv$
$\displaystyle d_{1}df+d_{3}d_{2}f-d_{1}^{2}w+d_{3}^{2}w-d_{1}d_{2}u+d_{3}du-
d_{1}d_{3}v-d_{3}d_{1}v.$

Subtracting the two equations and using the commutation relations (6.9) we get
$DD_{I_{1}}f=0$.

For the converse, observe that $DD_{I_{1}}f=0\Leftrightarrow
dd_{1}f=I_{2}dd_{1}f$. The $\partial\bar{\partial}$-lemma for $I_{2}$ gives
the existense of a smooth function $A_{1}$ such that $dd_{1}f=dd_{2}A_{1}$.
Similarly, using the Poincare lemma, we obtain
$d_{1}f-d_{2}A_{1}-dB_{1}=0,\quad d_{2}f-d_{3}A_{2}-dB_{2}=0,\quad
d_{3}f-d_{1}A_{3}-dB_{3}=0$ for a smooth functions
$A_{1},A_{2},A_{3},B_{1},B_{2},B_{3}$. The latter implies
$df+d_{1}(A_{2}+B_{1})+d_{2}(B_{2}-A_{3})+d_{3}(A_{1}-B_{3})=0.$ Set
$w=-A_{2}-B_{1},u=A_{3}-B_{2},v=B_{3}-A_{1}$ to get the equivalence between i)
and ii). ∎

####  6.2.1. Restriction on hyper-surfaces

In this section we shall denote with $\langle.,.\rangle$ the Euclidean scalar
product in $\mathbb{R}^{4n+4}\cong\mathbb{H}^{n+1}$ and with $\tilde{I}_{j}$,
$j=1,2,3$, the standard almost complex structures on $\mathbb{H}^{n+1}$. Let
$M$ be a smooth hyper-surface in $\mathbb{H}^{n+1}$ with a defining function
$\rho$, $M\ =\ \\{\rho=0\\}$, $d\rho\not=0$, and
$i:M\hookrightarrow\mathbb{H}^{n+1}$ be the embedding. It is not hard to see
that at every point $p\in M$ the subspace
$H_{p}=\bigcap_{j=1}^{3}\,\tilde{I}_{j}\,(T_{p}M)$ of the tangent space
$T_{p}M$ of $M$ at $p$ is the largest subspace invariant under the almost
complex structures and dim$H_{p}=4n$. We shall call $H_{p}$ the horizontal
space at $p$. Thus on the horizontal space $H$ the almost complex structures
$I_{j}$, $j=1,2,3$, are the restrictions of the standard almost complex
structures on $\mathbb{H}^{n+1}$. In particular, for a horizontal vector $X$
we have

(6.10) $\tilde{I}_{j}\,i_{*}X\ =\ i_{*}\,(I_{j}X).$

Let $\tilde{\theta}^{j}=\tilde{I}_{j}\ \frac{d\rho}{|d\rho|}$. We drop the
tilda in the notation of the almost complex structures when there is no
ambiguity.

We define three one-forms on $M$ by setting
$\theta^{j}=i^{*}\tilde{\theta}^{j}=i^{*}(\tilde{I}_{j}\
\frac{d\rho}{|d\rho|})$, i.e., $\theta_{j}\ (\ .\ )\ =\
-\frac{d\rho\,(\tilde{I}_{j}\,.\ )}{|d\rho|}\ =\ \langle\
.\,,\tilde{I}_{j}N\rangle,$ where $N=\frac{D\rho}{|D\rho|}$ is the unit normal
vector to $M$. We describe the hypersurfaces which inherit a natural
quaternionic contact structure from the standard structures on
$\mathbb{H}^{n+1}$ (see also [D1]) in the next

######  Proposition 6.12.

If $M$ is a smooth hypersurface of $\mathbb{H}^{n+1}$ then we have

(6.11)
$d\theta_{1}(I_{1}X,Y)=d\theta_{2}(I_{2}X,Y)=d\theta_{3}(I_{3}X,Y)\quad(X,\,Y\in
H)$

if and only if the restriction of the second fundamental form of $M$ to the
horizontal space is invariant with respect to the almost complex structures,
i.e. if $X$ and $Y$ are two horizontal vectors we have
$II(I_{j}X,I_{j}Y)=II(X,Y)$. Furthermore, if the restriction of the second
fundamental form of $M$ to the horizontal space is positive definite,
$II(X,X)>0$ for any non-zero horizontal vector $X$, then
$(M,\theta,I_{1},I_{2})$ is a quaternionic contact manifold.

###### Proof.

Let $D$ be the Levi-Civita connection on $\mathbb{R}^{4n+4}$ and $X,\ Y$ be
two horizontal vectors. As the horizontal space is the intersection of the
kernels of the one forms $\theta_{j}$ we have

(6.15) $d\theta_{1}(I_{1}X,Y)\ =\ -\theta_{1}([\tilde{I}_{1}X,Y])\ =\
-\langle[\tilde{I}_{1}X,Y],\tilde{I}_{1}N\rangle\\\ =\ -\langle
D_{\tilde{I}_{1}X}Y-D_{Y}(\tilde{I}_{1}X),\tilde{I}_{1}N\rangle\ =\ -\langle
D_{\tilde{I}_{1}X}Y,\tilde{I}_{1}N\rangle+\langle
D_{Y}(\tilde{I}_{1}X),\tilde{I}_{1}N\rangle\\\ =\ \langle
D_{\tilde{I}_{1}X}(\tilde{I}_{1}Y),N\rangle+\langle D_{Y}X,N\rangle\ =\
II(\tilde{I}_{1}X,\tilde{I}_{1}Y)+II(X,Y).$

Therefore $d\theta_{1}(I_{1}X,Y)=d\theta_{2}(I_{2}X,Y)$ iff
$II(I_{j}X,I_{j}Y)=II(X,Y)$.

The last claim of the proposition is clear from the above formula. In
particular, $g_{H}(X,Y)=II(X,Y)$ is a metric on the horizontal space when the
second fundamental form is positive definite on the horizontal space and we
have $d\theta_{1}(I_{1}X,Y)=2g_{H}(X,Y).$ Hence, $(M,\theta,I,J)$ becomes a
quaternionic contact structure. We denote the corresponding horizontal forms
with $\omega_{j}$, i.e., $\omega_{j}(X,Y)=g_{H}(I_{j}X,Y)$. ∎

Let us note also that in the situation as above
$g=g_{H}+\Sigma_{j=1}^{3}\theta_{j}\otimes\theta_{j}$ is a Riemannian metric
on $M$. In view of the above observations we define a QC-hypersurface of
$\mathbb{H}^{n+1}$ as follows.

######  Definition 6.13.

We say that a smooth embedded hypersurface of $\mathbb{H}^{n+1}$ is a _$QC$
-hypersurface_ if the restriction of the second fundamental form of $M$ to the
horizontal space is a definite symmetric form, which is invariant with respect
to the almost complex structures.

Clearly every sphere in $\mathbb{H}^{n+1}$ is a QC-hypersurface and this is
true also for the ellipsoids $\sum_{a}\frac{|q^{a}|^{2}}{b_{a}}=1$. In fact, a
hypersurface of $\mathbb{H}^{n+1}$ is a QC-hypersurface if and only if the
(Euclidean) Hessian of the defining function $\rho$, considered as a quadratic
form on the horizontal space, is a symmetric definite matrix from
$GL(n,\mathbb{H})$, the latter being the linear group of invertible matrices
which commute with the standard complex structures on $\mathbb{H}^{n}$. The
same statement holds for hypersurfaces in quaternionic Kähler and hyperkähler
manifolds.

######  Proposition 6.14.

Let $i:M\rightarrow\mathbb{H}^{n}$ be a QC hypersurface in $\mathbb{H}^{n}$,
$f$ a real-valued function on $M$. If $f=i^{*}F$ is the restriction to $M$ of
a Q-pluriharmonic function $F$ defined on $\mathbb{H}^{n}$, i.e. $F$ is the
real part of an anti-regular function $F+iW+jU+kV$, then:

(6.16) $\displaystyle
df=d(i^{*}F)=d_{1}(i^{*}W)+d_{2}(i^{*}U)+d(i^{*}V)\qquad\text{mod}\ \eta,$
(6.17) $\displaystyle DD_{I_{1}}f(X,I_{1}Y)\ =\ -4dF(D\rho)\,g_{H}(X,Y)\ -\
4(\xi_{2}f)\,\omega_{2}(X,Y)$

for any horizontal vector fields $X,Y\in H$.

###### Proof.

Let us prove first (6.16). Denote with small letters the restrictions of the
functions defined on $\mathbb{H}^{n}$. For $X\in H$ from (6.10) we have

$(i^{*}\,\tilde{I}_{1}\,dW)(X)=(\tilde{I}_{1}dW)(i_{*}\,X)=-dW(\tilde{I}_{1}\,i_{*}\,X)=-dW(i_{*}\,(I_{1}\
X))=-dw(I_{1}\,X)=d_{1}w\,(X).$

Applying the same argument to the functions $U$ and $V$ we see the validity of
(6.16).

Our goal is to write the equation for $f$ on $M$, using the fact that
$f=i^{*}F$. Let us consider the function $\lambda$, $\lambda\ =\
\frac{dF(D\rho)}{|D\rho|^{2}},$ and the one-form $d_{M}F$, $d_{M}F\ =\ dF\ -\
\lambda\,d\rho.$ Thus the one-form $df$ satisfies the equation $df\ =\
i^{*}(d_{M}F\ +\ \lambda\,d\rho)\ =\ i^{*}(d_{M}F),$ taking into account that
$(\lambda\circ i)\,d(\rho\circ i)\ =\ 0$ as $\rho$ is constant on $M$. From
Proposition 6.8, the assumption on $F$ is equivalent to
$DD_{\tilde{I}_{j}}F=0$. Therefore, we have

$\displaystyle 0\ =\ DD_{{\tilde{I}_{1}}}F\ =\ d{\tilde{I}_{1}}dF\ -\
{\tilde{I}_{2}}d{\tilde{I}_{1}}dF=\ d\,({\tilde{I}_{1}}d_{M}F\ +\
\lambda{\tilde{I}_{1}}d\rho)\ -\ {\tilde{I}_{2}}d\,({\tilde{I}_{1}}d_{M}F\ +\
\lambda{\tilde{I}_{1}}d\rho)$ $\displaystyle=\ d\,{\tilde{I}_{1}}d_{M}F\ +\
d\lambda\wedge{\tilde{I}_{1}}d\rho\ +\ \lambda d{\tilde{I}_{1}}d\rho-\
{\tilde{I}_{2}}d\,{\tilde{I}_{1}}d_{M}F\ -\
{\tilde{I}_{2}}(d\lambda\wedge{\tilde{I}_{1}}d\rho)\ -\
\lambda{\tilde{I}_{2}}{\tilde{I}_{1}}d\rho.$

Restricting to $M$, and in fact, to the horizontal space $H$ we find

(6.19) $\displaystyle 0\ =\ i^{*}\,(DD_{{\tilde{I}_{1}}}F)|_{H}=\ i^{*}d\
({\tilde{I}_{1}}d_{M}F)|_{H}\ +\ d(\lambda\circ i)\wedge
i^{*}({\tilde{I}_{1}}d\rho)|_{H}\ +\ (\lambda\circ
i)di^{*}({\tilde{I}_{1}}d\rho)|_{H}$ $\displaystyle\quad\quad-\
i^{*}({\tilde{I}_{2}}d\,{\tilde{I}_{1}}d_{M}F)|_{H}\ -\
i^{*}({\tilde{I}_{2}}(d\lambda\wedge{\tilde{I}_{1}}d\rho))|_{H}\ -\
(\lambda\circ i)\ i^{*}({\tilde{I}_{2}}d\,{\tilde{I}_{1}}d\rho)|_{H}.$

Since the horizontal space is in the kernel of the one-forms $\theta_{j}\ =\
\tilde{I}_{j}\,d\rho|_{H}$ it follows that
$i^{*}\,(\tilde{I}_{j}\,d\rho)|_{H}\ =\ 0.$ Hence, two of the terms in (6.19)
are equal to zero, and we have

$\displaystyle 0\ =\ i^{*}\,(DD_{{\tilde{I}_{1}}}F)_{|_{H}}\ =\
i^{*}(d\,{\tilde{I}_{1}}d_{M}F\ -\ {\tilde{I}_{2}}d\,{\tilde{I}_{1}}d_{M}F)\
_{|_{H}}+\ (\lambda\circ i)\ i^{*}\bigl{(}d\,{\tilde{I}_{1}}d\rho\ -\
{\tilde{I}_{2}}d\,{\tilde{I}_{1}}d\rho\bigr{)}_{|_{H}}.$

In other words for horizontal $X$ and $Y$ we have

(6.20) $i^{*}(d\,{\tilde{I}_{1}}d_{M}F\ -\
{\tilde{I}_{2}}d\,{\tilde{I}_{1}}d_{M}F)(X,IY)\ =\ -(\lambda\circ i)\
i^{*}\bigl{(}d\,{\tilde{I}_{1}}d\rho\ -\
{\tilde{I}_{2}}d\,{\tilde{I}_{1}}d\rho\bigr{)}(X,IY)$

The right-hand side is proportional to the metric. Indeed, recalling

$i^{*}\ (\tilde{I}_{j}\,d\,\rho)(X)\ =\ |d\rho|\
\theta_{j}(X)d\,\theta_{j}(X,Y)\ =\ 2g(I_{j}X,Y)$

we obtain the identity

$i^{*}\bigl{(}d\,{\tilde{I}_{1}}d\rho\ -\
{\tilde{I}_{2}}d\,{\tilde{I}_{1}}d\rho\bigr{)}(X,Y)\ =\ 2|d\rho|\ g(I_{1}X,Y)\
-\ 2|d\rho|\ g(I_{1}I_{2}X,I_{2}Y)\\\ =\ 2|d\rho|\ g(I_{1}X,Y)\ -\
2g(I_{3}X,I_{2}Y)\ =\ 4|d\rho|\ g(I_{1}X,Y).$

Let us consider now the term in the left-hand side of (6.20). Decomposing
$d_{M}F$ into horizontal and vertical parts we write $d_{M}F\ =\ d_{H}f\ +\
F_{j}\tilde{\theta}^{j}.$ ¿From the definitions of the forms
$\tilde{\theta}^{j}$ we have
$\tilde{I}_{1}\tilde{\theta}^{1}=\frac{d\rho}{|d\rho|},\quad\tilde{I}_{1}\tilde{\theta}^{2}=\tilde{\theta}^{3},\quad\tilde{I}_{1}\tilde{\theta}^{1}\tilde{\theta}^{3}=-\tilde{\theta}^{2}.$
Therefore

$d{\tilde{I}_{1}}d_{M}F\ =\ d{\tilde{I}_{1}}d_{H}F\ +\
dF_{j}\wedge{\tilde{I}_{1}}\theta^{j}\ +\ F_{1}d(\frac{d\rho}{|d\rho|})\ +\
F_{2}d\tilde{\theta}^{3}\ -\ F_{3}d\tilde{\theta}^{2}\\\ =\
d{\tilde{I}_{1}}d_{H}F\ +\ dF_{j}\wedge{\tilde{I}_{1}}\theta^{j}\ -\
|d\rho|^{-2}d|d\rho|\wedge d\rho\ +\ F_{2}d\tilde{\theta}^{3}\ -\
F_{3}d\tilde{\theta}^{2}\\\ {\tilde{I}_{2}}d{\tilde{I}_{1}}d_{M}F\ =\
{\tilde{I}_{2}}d{\tilde{I}_{1}}d_{H}F\ +\
{\tilde{I}_{2}}dF_{j}\wedge{\tilde{I}_{2}}{\tilde{I}_{1}}\theta^{j}\\\ -\
|d\rho|^{-2}{\tilde{I}_{2}}d|d\rho|\wedge{\tilde{I}_{2}}d\rho\ +\
F_{2}{\tilde{I}_{2}}d\tilde{\theta}^{3}\ -\
F_{3}{\tilde{I}_{2}}d\tilde{\theta}^{2}.$

From $I_{2}d\theta^{3}\ =\ -d\theta^{3}$, $I_{2}d\theta^{2}\ =\ d\theta^{2}$
and the above it follows

$i^{*}(d\,{\tilde{I}_{1}}d_{M}F-{\tilde{I}_{2}}d\,{\tilde{I}_{1}}d_{M}F)_{|_{H}}=DD_{I_{1}}f+F_{2}d\theta^{3}-F_{3}d\theta^{2}+F_{2}d\theta^{3}+F_{3}d\theta^{=}DD_{I_{1}}f+4F_{2}\
\omega_{3}.$

In conclusion, we proved $DD_{I_{1}}f(X,Y)\ =\ -4(\lambda\circ i)\
|\nabla\rho|\ g(I_{1}X,Y)\ -\ 4F_{2}\,\omega_{3}(X,Y)$ from where the claim of
the Proposition. ∎

###  6.3. Anti-CRF functions on Quaternionic contact manifold

Let $(M,\eta,\mathbb{Q})$ be a (4n+3)-dimensional quaternionic contact
manifold and $\nabla$ denote the Biquard connection on $M$. The equation
(6.16) suggests the following

######  Definition 6.15.

A smooth $\mathbb{H}$-valued function $F:M\longrightarrow\mathbb{H},\quad
F=f+iw+ju+kv,$ is said to be an _anti-CRF function_ if the smooth real valued
functions $f,w,u,v$ satisfy

(6.21) $df\ =\ d_{1}w\ +\ d_{2}u\ +\ d_{3}v\ \text{ mod }\ \eta,$

where $d_{i}\ =\ I_{i}\circ d.$

Choosing a local frame
$\\{T_{a},X_{a}=I_{1}T_{a},Y_{a}=I_{2}T_{a},Z_{a}=I_{3}T_{a},\xi_{1},\xi_{2},\xi_{3}\\},\quad
a=1,\dots,n$ it is easy to check that a $\mathbb{H}$-valued function
$F=f+iw+ju+kv$ is an anti-CRF function if it belongs to the kernel of the
operators

(6.22) $D_{T_{\alpha}}\ =\
T_{\alpha}-iX_{\alpha}-jY_{\alpha}-kZ_{\alpha},\qquad D_{T_{\alpha}}F\ =\
0,\qquad\alpha=1,\dots n.$

######  Remark 6.16.

We note that anti-CRF functions have different properties than the CRF
functions [Per1, Per2]which are defined to be in the kernel of the operator

$\overline{D}_{T_{\alpha}}\ =\
T_{\alpha}+iX_{\alpha}+jY_{\alpha}+kZ_{\alpha},\qquad\overline{D}_{T_{\alpha}}F\
=\ 0,\qquad\alpha=1,\dots n.$

Equation (6.21) and a small calculation give the following Proposition.

######  Proposition 6.17.

A $\mathbb{H}$-valued function $F=f+iw+ju+kv$ is an anti-CRF function if and
only if the smooth functions $f,w,u,v$ satisfy the horizontal Cauchy-Riemann-
Fueter equations

(6.25)
$\begin{array}[]{c}T_{\alpha}f=-X_{\alpha}w-Y_{\alpha}u-Z_{\alpha}v,\quad
X_{\alpha}f=T_{\alpha}w+Z_{\alpha}u-Y_{\alpha}v,\\\
Y_{\alpha}f=-Z_{\alpha}w+T_{\alpha}u+X_{\alpha}v,\quad
Z_{\alpha}f=Y_{\alpha}w-X_{\alpha}u+T_{\alpha}v.\end{array}$

Having the quaternionic contact form $\eta$ fixed, we may extend the
definitions (6.6) of $DD_{I_{i}}$ to the second order differential operator
$DD_{I_{i}}$ acting on the real-differentiable functions
$f:M\rightarrow\mathbb{R}$ by

(6.26)
$DD_{I_{i}}f:=dd_{i}f+d_{j}d_{k}f=dd_{i}f-I_{j}dd_{i}f=d(I_{i}df)-I_{j}(d(I_{i}df)).$

The following proposition provides some formulas, which shall be used later.

######  Proposition 6.18.

On a QC-manifold, for $X,Y\in H$, we have the following commutation relations

$\displaystyle
DD_{I_{i}}f(X,I_{i}Y)-DD_{I_{k}}f(X,I_{k}Y)=-I_{i}N_{I_{j}}(X,I_{i}Y)(f)-N_{I_{k}}(I_{j}X,I_{j}Y)(f),$
(6.27) $\displaystyle
d_{i}d_{j}f(X,Y)+d_{j}d_{i}f(X,Y)=-N_{I_{k}}(I_{j}X,I_{i}Y)(f),$
$\displaystyle dd_{i}f(X,Y)+d_{i}df(X,Y)=N_{I_{i}}(I_{i}X,Y)f,$ (6.28)
$\displaystyle
d_{i}^{2}f(X,Y)=-2\xi_{i}(f)\omega_{i}(X,Y)+2\xi_{j}(f)\omega_{j}(X,Y)+2\xi_{k}(f)\omega_{k}(X,Y),$

In particular, on a hyperhermitian contact manifold we have

$\displaystyle
DD_{I_{i}}f(X,I_{i}Y)-DD_{I_{k}}(X,I_{k}Y)=4\xi_{i}(f)\omega_{i}(X,Y)-4\xi_{j}(f)\omega_{j}(X,Y),$
(6.29) $\displaystyle
d_{i}d_{j}f(X,Y)+d_{j}d_{i}f(X,Y)=-4(\xi_{i}(f)\omega_{j}+\xi_{j}(f)\omega_{i}),$
$\displaystyle dd_{i}f+d_{i}df=4(\xi_{k}(f)\omega_{j}-\xi_{j}(f)\omega_{k}).$

###### Proof.

By the definition (6.26) we obtain the second and the third formulas in (6.27)
as well as $DD_{i}(X,Y)+(dd_{k}-d_{j}d_{i})(X,I_{j}Y)=-I_{i}N_{I_{j}}(X,Y)$.
The first equality in (6.27) is a consequence of the latter and the second
equality in (6.27). We have

$d_{i}^{2}f(X,Y)=-I_{i}d(I_{i}^{2}df)(X,Y)=d(df-\sum_{s=1}^{3}\xi_{s}(f)\eta_{s})(I_{i}X,I_{i}Y)$

which is exactly (6.28). If $H$ is formally integrable then the formula (4.33)
reduces to $N_{i}(X,Y)=T_{i}^{0,2}(X,Y)$. The equation (6.29) is an easy
consequences of the latter equality, (6.27) and (3.3) ∎

Let us make the conformal change $\bar{\eta}=\frac{1}{2h}\eta$. The
endomorphisms $\bar{I}_{i}$ will coincide with $I_{i}$ on the horizontal
distribution $H$ but they will have a different kernel - the new vertical
space $span\\{\bar{\xi}_{1},\bar{\xi}_{2},\bar{\xi}_{3}\\}$, where
$\bar{\xi}_{s}=2h\xi_{s}+I_{s}(\nabla h)$ (see (5.1)). Hence, for any
$P\in\Gamma(TM)$ we have

(6.32)
$\bar{I}_{i}(P)=\bar{I}_{i}(P-\sum_{s=1}^{3}\tilde{\eta}_{s}(P)\bar{\xi}_{s})=I_{i}(P-\frac{1}{2h}\sum_{s=1}^{3}\eta_{s}(P)(2h\xi_{s}+I_{s}(\nabla
h)))\\\ =I_{i}(P)+\frac{1}{2h}\\{\eta_{i}(P)\nabla h-\eta_{j}(P)I_{k}\nabla
h+\eta_{k}(P)I_{j}\nabla h\\}.$

######  Proposition 6.19.

Suppose $\bar{\eta}=\frac{1}{2h}\eta$ are two conformal to each other
structures.

a) The second order differential operator $DD_{I_{i}}$ (restricted on
functions) transforms as follows:

$DD_{\bar{I}_{i}}f-DD_{I_{i}}f=-2h^{-1}df(\nabla
h)\omega_{i}-2h^{-1}df(I_{j}\nabla h)\omega_{k}\qquad\text{mod}\ \eta.$

b) If $f$ is the real part of the anti-CRF function $f+iw+ju+kv$ then the two
forms

$\Omega_{i}\ =\ DD_{I_{i}}\,f\ -\ \lambda\,\omega_{i}\ +\
4(\xi_{j}f)\,\omega_{k}\qquad\text{mod}\ \eta$

are conformally invariant, where $\lambda\ =\
4\big{(}\xi_{1}w+\xi_{2}u+\xi_{3}v\big{)}.$

###### Proof.

a) For any $X,Y\in H$, we compute

$\displaystyle
d(\bar{I}_{i}df)(X,Y)=X(I_{i}df(Y))-Y(I_{i}df(X))-\bar{I}_{i}df[X,Y]=d(I_{i}df)(X,Y)+df(\bar{I}_{i}[X,Y]-I_{i}[X,Y])$

Here, we apply (6.32) to get

$d(\bar{I}_{i}df)(X,Y)=d(I_{i}df)(X,Y)+\frac{1}{h}\\{-df(\nabla
h)\omega_{i}(X,Y)+df(I_{k}\nabla h)\omega_{j}(X,Y)-df(I_{j}\nabla
h)\omega_{k}(X,Y)\\}.$

Now, apply the defining equation (6.26) to get the proof of a).

b) Assuming that $f$ is the real part of an anti-CRF function, from part a) we
have

$\displaystyle\bar{\Omega}_{i}\ -\ \Omega_{i}\ $ $\displaystyle=\
DD_{\bar{I}_{i}}\,f\ -\ DD_{I_{i}}\,f\ -\ \bar{\lambda}\,\omega_{i}\ +\
\lambda\,\omega_{i}\ +\ 4\bar{\xi}_{j}f\,\omega_{k}\ -\
4\xi_{j}f\,\omega_{k}\qquad\text{mod}\ \eta$
$\displaystyle=-4(\xi_{1}w+\xi_{2}u+\xi_{3}v)\omega_{i}-4\big{(}(I_{1}dh)w+(I_{2}dh)u+(I_{3}dh)v\big{)}\frac{\omega_{i}}{2h}$
$\displaystyle-\frac{2}{h}g(df,dh)\omega_{i}-\frac{2}{h}g(df,d_{j}h)\omega_{k}+4(\xi_{1}w+\xi_{2}u+\xi_{3}v)\omega_{i}+4g(df,d_{j}h)\frac{\omega_{k}}{2h}\
=\ 0\quad\text{mod}\ \eta,$

taking into account (6.21). ∎

We restrict our considerations to hyperhermitian contact manifolds.

######  Theorem 6.20.

If $f:M\rightarrow\mathbb{R}$ is the real part of an anti-CRF function
$f+iw+ju+kv$ on a (4n+3)-dimensional $(n>1)$ hyperhermitian contact manifold
$(M,\eta,Q)$.

Then the following equivalent conditions hold true:

  1. i)

The next equalities hold

(6.33) $DD_{I_{i}}f=\lambda\omega_{i}-4\xi_{j}(f)\omega_{k}\qquad\text{mod}\
\eta.$

  2. ii)

For any $X,Y\in H$ we have the equality

$\displaystyle(\nabla_{X}df)(Y)$
$\displaystyle+(\nabla_{I_{1}X}df)(I_{1}Y)+(\nabla_{I_{2}X}df)(I_{2}Y)+(\nabla_{I_{3}X}df)(I_{3}Y)$
$\displaystyle=\ 4\lambda
g(X,Y)+df(X)\alpha_{3}(I_{3}Y)+df(I_{1}X)\alpha_{3}(I_{2}Y)-df(I_{2}X)\alpha_{3}(I_{1}Y)-df(I_{3}X)\alpha_{3}(Y)$
$\displaystyle\quad+df(Y)\alpha_{3}(I_{3}X)+df(I_{1}Y)\alpha_{3}(I_{2}X)-df(I_{2}Y)\alpha_{3}(I_{1}X)-df(I_{3}Y)\alpha_{3}(X).$

  3. iii)

The function $f$ satisfies the second order differential equations

$\displaystyle\Re(D_{T_{\beta}}\overline{D}_{T_{\alpha}}f)=\lambda
g(T_{\beta},T_{\alpha})$ (6.34)
$\displaystyle\quad+df(\nabla_{T_{\beta}}T_{\alpha})+df(\nabla_{I_{1}T_{\beta}}I_{1}T_{\alpha})+df(\nabla_{I_{2}T_{\beta}}I_{2}T_{\alpha})+df(\nabla_{I_{3}T_{\beta}}I_{3}T_{\alpha})$
$\displaystyle\quad+df(T_{\beta})\alpha_{3}(I_{3}T_{\alpha})+df(I_{1}T_{\beta})\alpha_{3}(I_{2}T_{\alpha})-df(I_{2}T_{\beta})\alpha_{3}(I_{1}T_{\alpha})-df(I_{3}T_{\beta})\alpha_{3}(T_{\alpha})$
$\displaystyle\quad+df(T_{\alpha})\alpha_{3}(I_{3}T_{\beta})+df(I_{1}T_{\alpha})\alpha_{3}(I_{2}T_{\beta})-df(I_{2}T_{\alpha})\alpha_{3}I_{1}(T_{\beta})-df(I_{3}T_{\alpha})\alpha_{3}(T_{\beta})$
(6.35)
$\displaystyle\Re(iD_{T_{\beta}}\overline{D}_{T_{\alpha}}f)=\Re(D_{I_{1}T_{\beta}}\overline{D}_{T_{\alpha}}f),\quad\Re(jD_{T_{\beta}}\overline{D}_{T_{\alpha}}f)=\Re(D_{I_{2}T_{\beta}}\overline{D}_{T_{\alpha}}f),$
$\displaystyle\hskip
72.26999pt\Re(jD_{T_{\beta}}\overline{D}_{T_{\alpha}}f)=\Re(D_{I_{3}T_{\beta}}\overline{D}_{T_{\alpha}}f).$

The function $\lambda$ is determined by

(6.36) $\lambda\ =\ 4(\xi_{1}(w)+\xi_{2}(u)+\xi_{3}(v)).$

###### Proof.

The proof includes a number of steps and occupies the rest of the section.

_i)_ Suppose that there exists a smooth functions $w,u,v$ such that
$F=f+iw+ju+kv$ is an anti-CRF function. The defining equation (6.21) yields

(6.37) $df=d_{1}w+d_{2}u+d_{3}v+\sum_{s=1}^{3}\xi_{s}(f)\eta_{s},$

Since $d_{s}\eta_{t}(X,Y)=0$, for $s,t\in\\{1,2,3\\},X,Y\in H$, applying
(6.28) and (2.1), we obtain from (6.37)

$\displaystyle(dd_{1}f-dd_{3}u+dd_{2}v-2\xi_{1}(w)\omega_{1}-2\xi_{2}(w)\omega_{2}-2\xi_{3}(w)\omega_{3})(X,Y)=0,$
$\displaystyle(d_{1}df-
d_{1}d_{2}u-d_{1}d_{3}v+2\xi_{1}(w)\omega_{1}-2\xi_{2}(w)\omega_{2}-2\xi_{3}(w)\omega_{3})(X,Y)=0,$
$\displaystyle(d_{2}d_{3}f+d_{2}d_{1}u+d_{2}dv-2\xi_{1}(w)\omega_{1}+2\xi_{2}(w)\omega_{2}-2\xi_{3}(w)\omega_{3})(X,Y)=0,$
$\displaystyle(d_{3}d_{2}f+d_{3}du-
d_{3}d_{1}v+2\xi_{1}(w)\omega_{1}+2\xi_{2}(w)\omega_{2}-2\xi_{3}(w)\omega_{3})(X,Y)=0.$

Summing the first and the third equations, subtracting the second and the
fourth and using the commutation relations (6.29) we obtain (6.33) with the
condition (6.36) which proves $i)$.

$iii)$ Equations (6.22) and (6.25) yield

$\displaystyle 2\Re(D_{T_{\beta}}$ $\displaystyle\overline{D}_{T_{\alpha}}f)\
=\
2(T_{\beta}T_{\alpha}f+X_{\beta}X_{\alpha}f+Y_{\beta}Y_{\alpha}f+Z_{\beta}Z_{\alpha}f)$
$\displaystyle=(\Re(D_{T_{\beta}}\overline{D}_{T_{\alpha}}f)+\Re(D_{T_{\alpha}}\overline{D}_{T_{\beta}}f))+(\Re(D_{T_{\beta}}\overline{D}_{T_{\alpha}}f)-\Re(D_{T_{\alpha}}\overline{D}_{T_{\beta}}f))=$
$\displaystyle([T_{\beta},T_{\alpha}]+[X_{\beta},X_{\alpha}]+[Y_{\beta},Y_{\alpha}]+[Z_{\beta},Z_{\alpha}])f-([T_{\beta},X_{\alpha}]-[X_{\beta},T_{\alpha}]+[Y_{\beta},Z_{\alpha}]-[Z_{\beta},Y_{\alpha}])w$
$\displaystyle-([T_{\beta},Y_{\alpha}]-[X_{\beta},Z_{\alpha}]-[Y_{\beta},T_{\alpha}]+[Z_{\beta},X_{\alpha})u-([T_{\beta},Z_{\alpha}]+[X_{\beta},Y_{\alpha}]-[Y_{\beta},X_{\alpha}]-[Z_{\beta},T_{\alpha}])v.$

Expanding the commutators and applying (6.21), (3.3), (3.33) and (4.36) gives
(6.34). Similarly, one can check the validity of (6.35)

$i)\Leftrightarrow ii)\Leftrightarrow iii)$ The next lemma establishes the
equivalence between i), ii) and iii).

######  Lemma 6.21.

For any $X,Y\in H$ on a quaternionic contact manifold we have the identity

$DD_{I_{1}}f(X,I_{1}Y)=(\nabla_{X}df)Y+(\nabla_{I_{1}X}df)I_{1}Y+(\nabla_{I_{2}X}df)I_{2}Y+(\nabla_{I_{3}X}df)I_{3}X-4\xi_{2}(f)\omega_{2}(X,Y)\\\
-df(X)\alpha_{3}(I_{3}Y)+df(I_{1}X)\alpha_{2}(I_{3}Y)+df(I_{2}X)\alpha_{3}(I_{1}Y)-df(I_{3}X)\alpha_{2}(I_{1}Y)\\\
-df(Y)\alpha_{2}(I_{2}X)-df(I_{1}Y)\alpha_{3}(I_{2}X)+df(I_{2}Y)\alpha_{2}(X)+df(I_{3}Y)\alpha_{3}(X).$

###### Proof of Lemma 6.21.

Using the definition and also (3.33), (3.3) and (5.6) we derive the next
sequence of equalities

$(dd_{I_{1}}f)(X,Y)=-(\nabla_{X}df)(I_{1}Y)+(\nabla_{Y}df)(I_{1}X)-df(\nabla_{X}(I_{1}Y)-\nabla_{Y}(I_{1}X)-I_{1}[X,Y])\\\
=-(\nabla_{X}df)(I_{1}Y)+(\nabla_{Y}df)(I_{1}X)+\alpha_{2}(X)df(I_{3}Y)-\alpha_{3}(X)df(I_{2}Y)-\alpha_{2}(Y)df(I_{3}X)+\alpha_{3}(Y)df(I_{2}X)\\\
=\ -(\nabla_{X}df)I_{1}Y\ +\ (\nabla_{I_{1}X}df)Y\ -\ df(T(Y,\,I_{1}X))\\\
+\alpha_{2}(X)df(I_{3}Y)-\alpha_{3}(X)df(I_{2}Y)-\alpha_{2}(Y)df(I_{3}X)+\alpha_{3}(Y)df(I_{2}X).$
(6.49) $DD_{I_{1}}f(X,I_{1}Y)=(dd_{I_{1}}-I_{2}dd_{I_{1}})f\,(X,I_{1}Y)\\\
=(\nabla_{X}df)Y+(\nabla_{I_{1}X}df)I_{1}Y+(\nabla_{I_{2}X}df)I_{2}Y+(\nabla_{I_{3}X}df)I_{3}X-df(T(I_{1}Y,\,I_{1}X))-df(T(I_{3}Y,\,I_{3}X))\\\
-df(X)\alpha_{3}(I_{3}Y)+df(I_{1}X)\alpha_{2}(I_{3}Y)+df(I_{2}X)\alpha_{3}(I_{1}Y)-df(I_{3}X)\alpha_{2}(I_{1}Y)\\\
-df(Y)\alpha_{2}(I_{2}X)-df(I_{1}Y)\alpha_{3}(I_{2}X)+df(I_{2}Y)\alpha_{2}(X)+df(I_{3}Y)\alpha_{3}(X).$

A short calculation using (3.3) gives $\ df(T(I_{1}Y,\,I_{1}X))\ +\
df(T(I_{3}Y,\,I_{3}X))\ =4\xi_{2}(f)\omega_{2}(X,Y).$ Inserting the last
equality in (6.49) completes the proof of Lemma 6.21. ∎

Since the structure is hyperhermitian contact, with the help of (4.36) of
Lemma 6.21 the proof of Theorem 6.20 follows. ∎

We conjecture that the converse of the claim of Theorem 6.20 is true. At this
point we can prove Lemma 6.23, which supports the conjecture. First we prove a
useful technical result.

######  Lemma 6.22.

Suppose $M$ is a quaternionic contact manifold of dimension $(4n+3)>7$. If
$\psi$ is a smooth closed two-form whose restriction to $H$ vanishes, then
$\psi$ vanishes identically.

###### Proof of Lemma 6.22.

The hypothesis on $\psi$ show that $\psi$ is of the form $\psi\ =\
\sum_{s=1}^{3}\sigma_{s}\wedge\eta_{s},$ where $\sigma_{s}$ are 1-forms.
Taking the exterior differential and using (3.10), we obtain for $X\in H$

$\displaystyle
0=\sum_{a=1}^{4n}d\psi(e_{a},I_{i}e_{a},X)=-2\sum_{a=1}^{4n}\sum_{s=1}^{3}\sigma_{s}\wedge\omega_{s}(e_{a},I_{i}e_{a},X)=(4n-2)\sigma_{i}(X)+2\sigma_{j}(I_{k}X)-2\sigma_{k}(I_{j}X),$

where $e_{1},\dots,e_{4n}$ is an orthonormal basis on $H$. For $n>1$ the
latter implies $\sigma_{s_{|_{H}}}=0,s=1,2,3$. Hence, we have
$\psi=\sum_{1\leq s<t\leq}A_{st}\,\eta_{s}\wedge\eta_{t},$ where $A_{st}$ are
smooth functions on $M$. Now, the exterior derivative gives
$0=d\psi(e_{a},I_{s}e_{a},\xi_{t})\ =\ 2A_{st}.$ ∎

The assumption in the next Lemma is a kind of
$\partial\bar{\partial}_{H}$-lemma result, which we do not know how to prove
at the moment, but we believe that it is true. We show how it implies the
converse of Theorem 6.20.

######  Lemma 6.23.

Suppose, for $i=1,2,3$, $DD_{I_{i}}f\equiv
dd_{i}f+d_{j}d_{k}f=\sum_{s=1}^{3}p^{i}_{s}\omega_{s}\quad\text{mod}\ \eta$
implies

(6.50) $dd_{i}f-dd_{j}A_{i}\ =\
2\sum_{s=1}^{3}r^{i}_{s}\,\omega_{s}\qquad\text{mod}\ \eta$

for some function $A_{i}$ on a QC manifold of dimension $(4n+3)>7$. With this
assumption, if $DD_{I_{i}}f=\sum_{s=1}^{3}p^{i}_{s}\omega_{s}\quad\text{mod}\
\eta$, $i=1,2,3$, then $f$ is a real part of an anti-CRF-function.

###### Proof of Lemma 6.23.

Consider the closed 2-forms

$\Omega_{i}=d(d_{i}f-d_{j}A_{i}-\sum_{s=1}^{3}r^{i}_{s}\eta_{s}).$

We have $d\Omega_{i}=0$ and $\Omega_{i_{|_{H}}}=0$ due to (6.50) and (3.1).
Applying Lemma 6.22 we conclude $\Omega_{i}=0$, after which the Poincare lemma
yields

$d_{i}f-d_{j}A_{i}-dB_{i}=0\quad mod\quad\eta$

for some smooth functions $A_{1},A_{2},A_{3},B_{1},B_{2},B_{3}$. The latter
implies

$df+d_{1}(A_{2}+B_{1})+d_{2}(B_{2}-A_{3})+d_{3}(A_{1}-B_{3})=0\quad
mod\quad\eta.$

Setting $w=-A_{2}-B_{1},\quad u=A_{3}-B_{2},\quad v=B_{3}-A_{1}$ proves the
claim.∎

######  Corollary 6.24.

Let $f:M\rightarrow\mathbb{R}$ be a smooth real function on a
(4n+3)-dimensional $(n>1)$ 3-Sasakian manifold $(M,\eta)$. If $f$ is the real
part of an anti-CRF function $f+iw+ju+kv$ then:

  1. i)

The equation (6.33) holds.

  2. ii)

For any $X,Y\in H$ we have the equality

$\displaystyle(\nabla_{X}df)(Y)+(\nabla_{I_{1}X}df)(I_{1}Y)+(\nabla_{I_{2}X}df)(I_{2}Y)+(\nabla_{I_{3}X}df)(I_{3}Y)=4\lambda
g(X,Y).$

The function $\lambda$ is determined in (6.36).

######  Corollary 6.25.

Let $f:G(\mathbb{H})\rightarrow\mathbb{R}$ be a smooth real function on the
(4n+3)-dimensional $(n>1)$ quaternionic Heisenberg group endowed with the
standard flat quaternionic contact structure and
$\\{T_{a},X_{a},Y_{a},Z_{a},\quad a=1,\dots,4n\\}$ be $\nabla$-parallel basis
on $G(\mathbb{H})$. If $f$ is the real part of an anti-CRF function
$f+iw+ju+kv$ then the following equivalent conditions hold true:

  1. i)

The equation (6.33) holds.

  2. ii)

The horizontal Hessian of $f$ is given by

$\displaystyle T_{b}T_{a}f+X_{b}X_{a}f+Y_{b}Y_{a}f+Z_{a}Z_{b}=4\lambda
g(T_{b},T_{a});$

  3. iii)

The function $f$ satisfies the following second order differential equation

$D_{T_{b}}\overline{D}_{T_{a}}f=\lambda(g-i\omega_{1}-j\omega_{2}-k\omega_{3})(T_{b},T_{a});$

The function $\lambda$ is given by (6.36).

Proposition 6.2, Corollary 6.24 and Example 4.13 imply the next Corollary.

######  Corollary 6.26.

Let $(M,\eta)$ be a (4n+3)-dimensional $(n>1)$ 3-Sasakian manifold,
$f:M\rightarrow\mathbb{R}$ a positive smooth real function. Then the
conformally 3-Sasakian QC structrure $\bar{\eta}=f\eta$ is qc-pseudo Einstein
if and only if the operators $DD_{I_{s}}f,s=1,2,3$ satisfy (6.33). In
particular, if $f$ is real part of anti CRF function then the conformally
3-Sasakian qc structure $\bar{\eta}=f\eta$ is qc-pseudo Einstein.

##  7\. Infinitesimal Automorphisms

###  7.1. 3-contact manifolds

We start with the more general notion of 3-contact manifold $(M,H)$, where $H$
is an orientable codimension three distribution on $M$. Let $E\subset TM^{*}$
be the canonical bundle determined by $H$, i.e. the bundle of 1-forms with
kernel $H$. Hence, M is orientable if and only if $E$ is also orientable, i.e.
$E$ has a global non-vanishing section $vol_{E}$ locally given by $vol_{E}\ =\
\eta_{1}\wedge\eta_{2}\wedge\eta_{3}$. Denote by
$\eta=(\eta_{1},\eta_{2},\eta_{3})$ the local 1 -form with values in
$\mathbb{R}^{3}$. Clearly $H=Ker~{}\eta$.

######  Definition 7.1.

A $(4n+3)$-dimensional orientable smooth manifold $(M,\eta,H=Ker~{}\eta)$ is
said to be a _3-contact manifold_ if the restriction of each 2-form
$d\eta_{i},\ i=1,2,3$ to $H$ is non-degenerate, i.e.,

(7.1) $d\eta_{i}^{2n}\wedge\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\ =\ u_{i}\
vol_{M},\quad u_{i}>0,\quad i=1,2,3,$

and the following compatibility conditions hold

(7.2) $d\eta_{1}^{p}\wedge d\eta_{2}^{q}\wedge
d\eta_{3}^{r}\wedge\eta_{1}\wedge\eta_{2}\wedge\eta_{3}=0,\quad p+q+r=2n,\quad
0<p,q,r,<2n.$

Denote the restriction of $d\eta_{i}$ on $H$ by
$\Omega_{i},\quad\Omega_{i}=(d\eta)_{|_{H}},i=1,2,3$. The condition (7.1) is
equivalent to

$\Omega_{i}^{2n}\not=0,\quad
i=1,2,3,\qquad\Omega_{1}^{p}\wedge\Omega_{2}^{q}\wedge\Omega_{3}^{r}=0,\quad
p+q+r=2n,\quad 0<p,q,r<0.$

We remark that the notion of 3-contact structure is slightly more general than
the notion of QC structure. For example, any real hypersurface $M$ in
$\mathbb{H}^{n+1}$ with non-degenerate second fundamental form carries
3-contact structure defined in the beginning of Section 6.2.1 (cf. Proposition
6.12 and Definition 6.13 where this structure is QC if and only if (6.11)
holds, or equivalently, the second fundamental form is, in addition, invariant
with respect to the hypercomplex structure on $\mathbb{H}^{n+1}$). Another
examples of 3-contact structure is the so called quaternionic CR structure
introduced in [KN] and the so called weak QC structures considered in [D1].
Note that in these examples the 1-form $\eta=(\eta_{1},\eta_{2},\eta_{3})$ are
globally defined.

On any 3-contact manifold $(M,\eta,H)$ there exists a unique triple
$(\xi_{1},\xi_{2},\xi_{3})$ of vector fields transversal to $H$ determined by
the conditions

$\eta_{i}(\xi_{j})=\delta_{ij},\quad(\xi_{i}\lrcorner d\eta_{i})_{|_{H}}=0.$

We refer to such a triple as fundamental vector fields or Reeb vector fields
and denote $V=span\\{\xi_{1},\xi_{2},\xi_{3}\\}$. Hence, we have the splitting
$TM=H\oplus V$.

The 3-contact structure $(\eta,H)$ and the vertical space $V$ are determined
up to an action of $GL(3,\mathbb{R})$, namely for any $GL(3,\mathbb{R})$
matrix $\Phi$ with smooth entries the structure $\Phi\cdot\eta$ is again a
3-contact structure. Indeed, it is an easy algebraic fact that the conditions
(7.1) and (7.2) also hold for $\Phi\cdot\eta$. The Reeb vector field are
transformed with the matrix with entries the adjunction quantities of $\Phi$,
i.e. with the inverse matrix $\Phi^{-1}$. This leads to the next

######  Definition 7.2.

A diffeomorphism $\phi$ of a 3-contact manifold $(M,\eta,H)$ is called a
_3-contact automorphism_ if $\phi$ preserves the 3-contact structure $\eta$,
i.e.,

(7.3) $\phi^{*}\eta=\Phi\cdot\eta,$

for some matrix $\Phi\in GL(3,\mathbb{R})$ with smooth functions as entries
and $\eta=(\eta_{1},\eta_{2},\eta_{3})^{t}$ is considered as an element of
$\mathbb{R}^{3}$.

The infinitesimal versions of these notions lead to the following definition.

######  Definition 7.3.

A vector field $Q$ on a 3-contact manifold $(M,\eta,H)$ is an infinitesimal
generator of a 3-contact automorphism (3-contact vector field) if its flow
preserves the 3-contact structure, i.e.

$\mathcal{L}_{Q}\,\eta=\phi\cdot\eta,\qquad\phi\in gl(3,\mathbb{R}).$

We show that any 3-contact vector field on a 3-contact manifold depend on
3-functions which satisfy certain differential relations. We begin with
describing infinitesimal automorfisms of the 3-contact structure $\eta$ i.e.
vector field $Q$ whose flow satisfies (7.3). Our main observation is that
3-contact vector fields on a 3-contact manifold are completely determined by
their vertical components in the sense of the following

######  Proposition 7.4.

Let ($M,\eta,H)$ be a 3-contact manifold. A smoth vector field $Q$ on $M$ is
3-contact vector field if and only if the functions
$f_{i}=\eta_{i}(Q),i=1,2,3$ satisfy the next compatibility conditions on $H$

(7.4) $\displaystyle u_{i}(df_{i}+f_{j}(\xi_{j}\lrcorner
d\eta_{i})+f_{k}(\xi_{k}\lrcorner
d\eta_{i}))_{|_{H}}\wedge\Omega_{i}^{(2n-1)})=$ $\displaystyle
u_{j}(df_{j}+f_{k}(\xi_{k}\lrcorner d\eta_{j})+f_{i}(\xi_{i}\lrcorner
d\eta_{j}))_{|_{H}}\wedge\Omega_{j}^{(2n-1)})\quad{\text{o}n}\quad H.$

The 3-contact vector field $Q$ has the form

$Q=Q_{h}+f_{1}\xi_{1}+f_{2}\xi_{2}+f_{3}\xi_{3},$

where $Q_{h}$ is the horizontal 3-contact hamiltonian field of
$(f_{1},f_{2},f_{3})$ defined on $H$ by

(7.5) $Q_{h}\lrcorner\eta_{i}=0,\quad
Q_{h}\lrcorner(\Omega_{i})=-df_{i}-f_{j}(\xi_{j}\lrcorner
d\eta_{i})-f_{k}(\xi_{k}\lrcorner d\eta_{i}),\quad
i=1,2,3,\quad{\text{o}n}\quad H.$

###### Proof.

For a vector field $Q\in\Gamma(TM)$ we write
$Q=Q_{H}+\sum_{s=1}^{3}\eta_{s}(Q)\xi_{s}$ where $Q_{H}\in H$ is the
horizontal part of $Q$. Applying (3.10), we calculate

(7.6) $\displaystyle\mathcal{L}_{Q}\,\eta_{i}=Q\lrcorner
d\eta_{i}+d(Q\lrcorner\eta_{i})=$ $\displaystyle
Q_{H}\lrcorner\Omega_{i}+[d(\eta_{i}(Q))+\eta_{i}(Q)\xi_{i}\lrcorner
d\eta_{i}+\eta_{j}(Q)\xi_{j}\lrcorner d\eta_{i}+\eta_{k}(Q)\xi_{k}\lrcorner
d\eta_{i}]_{|_{H}}$
$\displaystyle+[\xi_{i}(\eta_{i}(Q))-\eta_{j}(Q)d\eta_{i}(\xi_{i},\xi_{j})-\eta_{k}(Q)d\eta_{i}(\xi_{i},\xi_{k})]\eta_{i}$
$\displaystyle+[\xi_{j}(\eta_{i}(Q))+d\eta_{i}(Q,\xi_{j})]\eta_{j}+[\xi_{k}(\eta_{i}(Q))+d\eta_{i}(Q,\xi_{k})]\eta_{k}.$

Suppose $Q$ is a 3-contact vector field. Then (7.6) and the compatibility
conditions (7.2) imply that $f_{i}$ and $Q_{H}$ necessarily satisfy (7.4) and
(7.5), respectively. Therefore $Q_{H}=Q_{h}$. The converse follows from (7.6)
and the conditions of the proposition. ∎

The last Proposition implies that the space of 3-contact vector fields is
isomorphic to the space of triples consisting of smooth function
$f_{1},f_{2},f_{3}$ satisfying the compatibility conditions (7.4).

######  Corollary 7.5.

Let $(M,\eta)$ be a 3-contact manifold. Then

$\xi_{i}\lrcorner d\eta_{j}{|_{H}}=0,\quad i,j=1,2,3.$

###  7.2. QC vector fields

Suppose $(M,g,\mathbb{Q})$ is a quaternionic contact manifold.

######  Definition 7.6.

A diffeomorphism $\phi$ of a QC manifold $(M,[g],\mathbb{Q})$ is called a
_conformal quaternionic contact automorphism (conformal qc-automorphism)_ if
$\phi$ preserves the QC structure, i.e.

$\phi^{*}\eta=\mu\Psi\cdot\eta,$

for some positive smooth function $\mu$ and some matrix $\Psi\in SO(3)$ with
smooth functions as entries and $\eta=(\eta_{1},\eta_{2},\eta_{3})^{t}$ is a
local 1-form considered as an element of $\mathbb{R}^{3}$.

In view of the uniqueness of the possible associated almost complex
structures, a quaternionic contact automorpism will preserve also the
associated (if any) almost complex structures, $\phi^{*}\mathbb{Q}=\mathbb{Q}$
and consequently, it will preserve the conformal class [g] on $H$. We note
that conformal QC diffeomorphisms on $S^{4n+3}$ are considered in [Kam]. The
infinitesimal versions of these notions lead to the following definition.

######  Definition 7.7.

A vector field $Q$ on a QC manifold $(M,[g],\mathbb{Q})$ is an infinitesimal
generator of a conformal quaternionic contact automorphism (_QC vector field_
for short) if its flow preserves the QC structure, i.e.

(7.7) $\mathcal{L}_{Q}\,\eta=(\nu I+O)\cdot\eta,$

where $\nu$ is a smooth function and $O\in so(3)$.

In view of the discussion above a QC vector field on a QC manifold
$(M,\eta,\mathbb{Q})$ satisfies the conditions.

(7.8) $\displaystyle\mathcal{L}_{Q}\,g=\nu g,$ (7.9)
$\displaystyle\mathcal{L}_{Q}\,I=O\cdot I,\qquad O\in so(3),\quad
I=(I_{1},I_{2},I_{3})^{t},$

If the flow of a vector field $Q$ is a conformal diffeomorphism of the
horizontal metric $g$, i.e. (7.8) holds, we shall call it _infinitesimal
conformal isometry_. If the function $\nu=0$ then $Q$ is said to be
_infinitesimal isometry_.

A QC vector field on a QC manifold is a 3-contact vector field of special
type. Indeed, let $\sharp$ be the musical isomorphism between $T^{*}M$ and
$TM$ with respect to the fixed Riemannian metric $g$ on $TM$ and recall that
the forms $\alpha_{j}$ were defined in (3.34). We have

######  Proposition 7.8.

Let $(M,g,\mathbb{Q})$ be a quaternionic contact manifold. The vector field
$Q$ is a QC vector field if and only if

(7.10) $Q\ =\ \frac{1}{2}\big{(}f_{j}\,I_{i}\,\alpha^{\sharp}_{k}\ -\
f_{k}\,I_{i}\,\alpha^{\sharp}_{j}\ -\ I_{i}\,(df_{i})^{\sharp}\big{)}\ +\
\sum_{s=1}^{3}f_{s}\,\xi_{s},$

for some functions $f_{1},f_{2}$ and $f_{3}$ such that for any positive
permutation $(i,j,k)$ of $(1,2,3)$ we have

(7.11) $\displaystyle f_{j}\,d\eta_{i}(\xi_{j},\xi_{i})+\
f_{k}\,d\eta_{i}(\xi_{k},\xi_{i})\ +\ \xi_{i}\,f_{i}\ =\
f_{k}\,d\eta_{j}(\xi_{k},\xi_{j})+\ f_{i}\,d\eta_{j}(\xi_{i},\xi_{j})\ +\
\xi_{j}\,f_{j}\,$ (7.12) $\displaystyle f_{i}\,d\eta_{i}(\xi_{i},\xi_{j})+\
f_{k}\,d\eta_{i}(\xi_{k},\xi_{j})\ +\ \xi_{j}\,f_{i}=\ -\
f_{j}\,d\eta_{j}(\xi_{j},\xi_{i})\ -\ f_{k}\,d\eta_{j}(\xi_{k},\xi_{i})\ -\
\xi_{i}\,f_{j},$ (7.13) $\displaystyle f_{j}\,I_{i}(\alpha_{k})^{\sharp}\ -\
f_{k}\,I_{i}(\alpha_{j})^{\sharp}\ -\ I_{i}(df_{i})^{\sharp}\ =\
f_{i}\,I_{k}(\alpha_{j})^{\sharp}\ -\ f_{j}\,I_{k}(\alpha_{i})^{\sharp}\ -\
I_{k}(df_{k})^{\sharp}$

###### Proof.

Notice that (7.10) implies $f_{i}=\eta_{i}(Q)$. By Cartan’s formula (7.7) is
equivalent to

$Q\lrcorner d\eta_{i}\ +\ df_{i}\ =\ \nu\,\eta_{i}\ +\ o_{is}\,\eta_{s}.$

In other words, both sides must be the same when evaluated on $\xi_{t}$,
$t=1,2,3$ and also when restricted to the horizontal bundle. Let
$Q=Q_{H}+\sum_{s=1}^{3}f_{s}\xi_{s}$. Consider first the action on the
vertical vector fields. Pairing with $\xi_{t}$ and taking successively
$t=i,j,k$ gives

$\displaystyle f_{j}\,d\eta_{i}(\xi_{j},\xi_{i})+\
f_{k}\,d\eta_{i}(\xi_{k},\xi_{i})\ +\ \xi_{i}\,f_{i}\ =\ \nu\ +\ o_{ii}$
(7.14) $\displaystyle\alpha_{k}(Q_{H})\ +\ f_{i}\,d\eta_{i}(\xi_{i},\xi_{j})+\
f_{k}\,d\eta_{i}(\xi_{k},\xi_{j})\ +\ \xi_{j}\,f_{i}\ =\ o_{ij}$
$\displaystyle-\alpha_{j}(Q_{H})\ +\ f_{i}\,d\eta_{i}(\xi_{i},\xi_{k})+\
f_{j}\,d\eta_{i}(\xi_{j},\xi_{k})\ +\ \xi_{k}\,f_{i}\ =\ o_{ik}.$

Equating the restrictions to the horizontal bundle, i.e.,
$d\eta_{i}(Q,.)|_{H}\ +\ df_{i}|_{H}\ =\ 0$, gives

$\Big{(}f_{j}\,d\eta_{i}(\xi_{j},.)\ +\ f_{k}\,d\eta_{i}(\xi_{k},.)\ +\
d\eta_{i}(Q_{H},.)\ +\ df_{i}\Big{)}|_{H}\ =\ 0.$

Since $g(A,.)|_{H}\ =\ 0\Leftrightarrow A\ =\
\sum_{s=1}^{3}\eta_{s}(A)\xi_{s}$, the last equation is equivalent to

(7.15) $-f_{j}\alpha^{\sharp}_{k}\ +\ f_{k}\alpha^{\sharp}_{j}\ -\
2I_{i}Q_{H}\ +\ (df_{i})^{\sharp}\ =\
\sum_{s=1}^{3}\big{(}-f_{j}\alpha_{k}(\xi_{s})\ +\ f_{k}\alpha_{j}(\xi_{s})\
+\ \xi_{s}f_{i}\big{)}\,\xi_{s}.$

Acting with $I_{i}$ determines $2Q_{H}\ =\ f_{j}\,I_{i}(\alpha_{k})^{\sharp}\
-\ f_{k}\,I_{i}(\alpha_{j})^{\sharp}\ -\ I_{i}(df_{i})^{\sharp}$, which
implies (7.10). In addition we have

$\alpha_{j}(Q_{H})\ =\
\frac{1}{2}\Big{(}f_{j}\,\alpha_{j}(I_{i}(\alpha_{k})^{\sharp})\ -\
f_{k}\,\alpha_{j}(I_{i}(\alpha_{j})^{\sharp})\ -\
\alpha_{j}(I_{i}(df_{i})^{\sharp})\Big{)}$

On the other hand, $o\in so(3)$ is equivalent to $o$ being a skew symmetric
which is equivalent to (7.11) and (7.12), by the above computations.
Therefore, if we are given three functions $f_{1},\ f_{2},\ f_{3}$ satisfying
(7.11), (7.12) and (7.13), then we define $Q$ by (7.10). Using (7.14) we
define $\nu$ and $o$ with $o\in so(3)$. With these definitions $Q$ is a QC
vector field. ∎

Using the formulas in Example 4.14 we obtain from Proposition 7.8 the
following ’3-hamiltonian’ form of a QC vector field on 3-Sasakian manifold.

######  Corollary 7.9.

Let $(M,\eta)$ be a 3-Sasakian manifold. Then any QC vector field $Q$ has the
form

$Q=Q_{h}+f_{1}\xi_{1}+f_{2}\xi_{2}+f_{3}\xi_{3},$

where the smooth functions $f_{1},f_{2},f_{3}$ satisfy the conditions

$d_{i}f_{i}=d_{j}f_{j},\quad\xi_{i}(f_{i})=\xi_{j}(f_{j}),\quad\xi_{i}(f_{j})=-\xi_{j}(f_{i}),\quad
i,j=1,2,3,$

and the horizontal part $Q_{h}\in H$ is determined by

$Q_{h}\lrcorner d\eta_{i}=d_{i}f_{i},\quad i\in\\{1,2,3\\}.$

The matrix in (7.7) has the form

$\displaystyle\nu
I_{d_{3}}+O=\left(\begin{array}[]{ccc}\xi_{1}(\eta_{1}(Q))&-\xi_{1}(\eta_{2}(Q))-2\eta_{3}(Q)&-\xi_{1}(\eta_{3}(Q))+2\eta_{2}(Q)\\\
\xi_{1}(\eta_{2}(Q))+2\eta_{3}(Q)&\xi_{1}(\eta_{1}(Q))&-\xi_{2}(\eta_{3}(Q))-2\eta_{1}(Q)\\\
\xi_{1}(\eta_{3}(Q))-2\eta_{2}(Q)&\xi_{2}(\eta_{3}(Q))+2\eta_{1}(Q)&\xi_{1}(\eta_{1}(Q))\end{array}\right).$

In particular, the Reeb vector fields $\xi_{1},\xi_{2},\xi_{3}$ are 3-contact
vector fields.

Corollary 7.5 tells us that on a QC manifold the Reeb vector fields
$\xi_{1},\xi_{2},\xi_{3}$ are 3-contact exactly when the connection 1-forms
vanish on $H$. This combined with Corollary 4.18 gives 3-Sasakian structure
compatible with the given 3-contact structure $H$, if the qc-scalar curvature
is not zero (see Corollary 7.14 below).

First we shall investigate some useful properties of a QC vector field.

######  Proposition 7.10.

Let $(M,[g],\mathbb{Q})$ be QC manifold and $Q$ be a QC vector field
determined by (7.7) and (7.14). The next equality hold

$d\eta_{i}([Q,I_{i}X]^{\perp},Y)+d\eta_{i}(I_{i}X,[Q,Y]^{\perp})=0$

###### Proof.

We have using (7.7) that

(7.16)
$\mathcal{L}_{Q}\,d\eta_{i}(I_{i}X,Y)=2(\mathcal{L}_{Q}\,\omega_{i})(I_{i}X,Y)-d\eta_{i}([Q,I_{i}X]^{\perp},Y)-d\eta_{i}(I_{i}X,[Q,Y]^{\perp})\\\
=-2(\mathcal{L}_{Q}\,g)(X,Y)+2g((\mathcal{L}_{Q}\,I_{i})I_{i}X,Y)-d\eta_{i}([Q,I_{i}X]^{\perp},Y)-d\eta_{i}(I_{i}X,[Q,Y]^{\perp})\\\
=(d\mathcal{L}_{Q}\,\eta_{i})(I_{i}X,Y)=(d\nu\wedge\eta_{i}+\nu
d\eta_{i}+do_{ij}\wedge\eta_{j}+o_{ij}d\eta_{j}+do_{ik}\wedge\eta_{k}+o_{ik}d\eta_{k})(I_{i}X,Y)\\\
=-2\nu g(X,Y)-2o_{ij}\omega_{k}(X,Y)+2o_{ik}\omega_{j}(X,Y),$

where $o_{st}$ are the entries of the matrix $O$ given by (7.14). Apply (7.8)
and (7.9) to (7.16) to get the assertion. ∎

We are going to characterize the vanishing of the torsion of the Biquard
connection in terms of the existence of some special vertical vector fields.
More precisely, we have the following Theorem.

######  Theorem 7.11.

Let $(M,g,\mathbb{Q})$ be a QC manifold with non zero qc-scalar curvature. The
following conditions are equivalent

  1. i)

Each Reeb vector field is a QC vector field;

  2. ii)

The torsion of the Biquard connection is identically zero;

  3. iii)

Each Reeb vector field preserves the horizontal metric and the quaternionic
structure simultaneously, i.e. (7.8) with $\nu=0$ and (7.9) hold for
$Q=\xi_{i},i=1,2,3$;

  4. iv)

There exists a local 3-Sasakian structure in the sense of Theorem 1.3

###### Proof.

In the course of the proof we shall prove two Lemmas of independent interest.
Given a vector field $Q$, we define the symmetric tensor $T^{0}_{Q}$ and the
skew-symmetric tensor $u_{Q}$

(7.17) $T^{0}_{Q}\ =\ \sum_{s=1}^{3}\eta_{s}(Q)\,T^{0}_{\xi_{s}},\qquad u_{Q}\
=\ \sum_{s=1}^{3}\eta_{s}(Q)\,I_{s}u,$

respectively, such that, $T(Q,X,Y)\ =\ g(T^{0}_{Q}X,Y)\ +\ g(u_{Q}X,Y),$

######  Lemma 7.12.

The tensors $T^{0}_{Q}$ and $u_{Q}$ lie in the $[-1]$ component associated to
the operator ${\dagger}$ cf. (2.3) and (2.2).

###### Proof of Lemma 7.12.

By the definition of $u_{Q}$, we have $g(u_{Q}I_{1}X,I_{1}Y)\ =\
\sum_{s=1}^{3}\eta_{s}(Q)\,g(I_{s}uX,Y)\ $ and after summing we find

$\sum_{j=1}^{3}g(u_{Q}I_{j}X,I_{j}Y)\
=\sum_{j=1}^{3}\\-\eta_{j}(Q)\,g(I_{j}uX,Y)\ =\ -g(u_{Q}X,Y).$

We turn to the second claim. Recall that $T^{0}_{\xi_{j}}$ anti-commutes with
$I_{j}$, see (2.8). Hence,

(7.21) $g(T^{0}_{Q}I_{1}X,I_{1}Y)\ =\ -\eta_{1}(Q)\,g(T^{0}_{\xi_{1}}X,Y)-\
\eta_{2}(Q)\,[g(T^{0\ --+}_{\xi_{2}}X,Y)\ -g(T^{0\ +--}_{\xi_{2}}X,Y)]\\\ -\
\eta_{3}(Q)\,[g(T^{0\ -+-}_{\xi_{2}}X,Y)\ -g(T^{0\ +--}_{\xi_{3}}X,Y)],\\\
g(T^{0}_{Q}I_{2}X,I_{2}Y)\ =\ -\eta_{2}(Q)\,g(T^{0}_{\xi_{2}}X,Y)-\
\eta_{1}(Q)\,[g(T^{0\ --+}_{\xi_{1}}X,Y)\ -g(T^{0\ -+-}_{\xi_{1}}X,Y)]\\\ -\
\eta_{3}(Q)\,[g(T^{0\ +--}_{\xi_{3}}X,Y)\ -g(T^{0\ -+-}_{\xi_{3}}X,Y)],\\\
g(T^{0}_{Q}I_{3}X,I_{3}Y)\ =\ -\eta_{3}(Q)\,g(T^{0}_{\xi_{3}}X,Y)-\
\eta_{1}(Q)\,[g(T^{0\ -+-}_{\xi_{1}}X,Y)\ -g(T^{0\ --+}_{\xi_{2}}X,Y)]\\\ -\
\eta_{2}(Q)\,[g(T^{0\ +--}_{\xi_{2}}X,Y)\ -g(T^{0\ --+}_{\xi_{2}}X,Y)].$

Summing the above three equations we come to

$\sum_{j=1}^{3}g(T^{0}_{Q}I_{j}X,I_{j}Y)\ =\
-\sum_{j=1}^{3}g(Q,\xi_{j})\,g(T^{0}_{\xi_{j}}X,Y)\ =\ -g(T^{0}_{Q}X,Y),$

which finishes the proof of Lemma 7.12. ∎

######  Lemma 7.13.

If $Q$ is an infinitesimal conformal isometry whose flow preserves the
quaternionic structure then the next two equalities hold

(7.22) $\displaystyle g(\nabla_{X}Q,Y)\ +\ g(\nabla_{Y}Q,X)\ +\
2g(T^{0}_{Q}X,Y)=\nu\,g(X,Y),$ (7.23) $\displaystyle
3g(\nabla_{X}Q,Y)-\sum_{s=1}^{3}g(\nabla_{I_{s}X}Q,I_{s}Y)+4g(T^{0}_{Q}X,Y)+4g(u_{Q}X,Y)=-2\sum_{(ijk)}L_{ij}(Q)\omega_{k}(X,Y),$

where the sum is over all even permutation of (1,2,3) and

(7.26)
$L_{ij}(Q)=-L_{ji}(Q)=\xi_{j}(\eta_{i}(Q))-\eta_{j}(Q)d\eta_{j}(\xi_{i},\xi_{j})\\\
+\frac{1}{2}\eta_{k}(Q)\left(\frac{Scal}{8n(n+2)}+d\eta_{j}(\xi_{k},\xi_{i})-d\eta_{i}(\xi_{j},\xi_{k})-d\eta_{k}(\xi_{i},\xi_{j})\right).$

###### Proof of Lemma 7.13.

In terms of the Biquard connection (7.8) reads exactly as (7.22). Furthermore,
from (7.9), (7.14) and (3.33) it follows

(7.29) $o_{ij}I_{j}X+o_{ik}I_{k}X=\ (\mathcal{L}_{Q}\,I_{i})(X)=\\\ =\
-\nabla_{I_{i}X}Q\ +I_{i}\nabla_{X}Q-\alpha_{j}(Q)I_{k}X+\alpha_{k}(Q)I_{j}X-\
T(Q,I_{i}X)\ +\ I_{i}T(Q,X).$

A use of (7.14), (3.34) and (3.35) allows us to write the last equation in the
form

(7.32) $g(\nabla_{X}Q,Y)\ -\ g(\nabla_{I_{i}X}Q,I_{i}Y)+\ T(Q,X,Y)\ -\
T(Q,I_{i}X,I_{i}Y)\ \\\
=(o_{ij}-\alpha_{k}(Q))\omega_{k}(X,Y)-(o_{ik}+\alpha_{j}(Q))\omega_{j}(X,Y)\\\
=-L_{ij}(Q)\omega_{k}(X,Y)+L_{ik}(Q)\omega_{j}(X,Y),$

where $L_{ij}(Q)$ satisfy (7.26). Summing the above identities for the three
almost complex structures and applying Lemma 7.12, we obtain (7.23), which
completes the proof of Lemma 7.13. ∎

We are ready to finish the proof of Theorem 7.11. Let $\xi_{i},i=1,2,3$ be QC
vector fields. Then (7.22) for $Q=\xi_{i}$ yields
$T_{\xi_{i}}=0,i=1,2,3,\quad\nu=0$ since $T_{\xi_{i}}$ is trace-free.
Consequently, for any cyclic permutation $(i,j,k)$ of $(1,2,3)$, (7.23) and
(7.26) imply

$u_{\xi_{i}}=0,\quad d\eta_{j}(\xi_{i},\xi_{j})=0,\quad
d\eta_{i}(\xi_{j},\xi_{k})=\frac{Scal}{8n(n+2)}$

by comparing the trace and the trace-free part. Hence ii) follow.

Conversely, if the torsion of the Biquard connection vanishes, then (7.22) is
trivially satisfied for $\nu=0$ and (7.29) yields (7.9) with
$o_{ij}=\alpha_{k}$. This establishes the equivalence between ii) and iii).

The other equivalences in the theorem follow from Theorem 1.3, Example 4.13,
Corollary 7.5 and Corollary 4.18. ∎

######  Corollary 7.14.

Let $(M,g,\mathbb{Q})$ be a QC manifold with non zero qc-scalar curvature. The
following conditions are equivalent

  1. i)

There exists a local 3-Sasakian structure compatible with $H=Ker~{}\eta$;

  2. ii)

There are three linearly independent transversal QC-vector fields.

###### Proof.

Let $\gamma_{1},\gamma_{2},\gamma_{3}$ be linearly independent transversal QC-
vector fields. Then there exist 1-forms
$\eta_{\gamma_{1}},\eta_{\gamma_{2}},\eta_{\gamma_{3}}$ satisfying
$\eta_{\gamma_{i}}(\gamma_{j})=\delta_{ij}$, where $\delta_{ij}$ is the
Kroneker symbol. In view of the proof of Theorem 7.11 it is sufficient to show
$\gamma_{1},\gamma_{2},\gamma_{3}$ are the Reeb vector field for
$\eta_{\gamma}$, i.e. we have to show that the compatibility conditions (2.5)
are satisfied. Indeed, the fact that $\gamma_{i},i=1,2,3$ are QC vector fields
means that (7.7) hold with respect to $\eta_{\gamma}$. Then (7.14) gives
$\nu=0$ and the second line of (7.6), for $\eta_{\gamma}$ and
$Q=\gamma_{i},i=1,2,3$, imply (2.5) for the structure $\eta_{\gamma}$. Theorem
7.11 completes the proof. ∎

In the particular case when the vector field $Q$ is the gradient of a function
defined on the manifold $M$, we have the following formulas.

######  Corollary 7.15.

If $h$ is a smooth real valued function on $M$ and $Q\ =\ \nabla h$ is a QC
vector field, then for any horizontal vector fields $X$ and $Y$ we have

  1. a)

$[\ (\nabla dh)]_{[3][0]}(X,Y)\ =\ 0$

  2. b)

$[\ \nabla dh\ ]_{[sym][-1]}(X,Y)\ =\ -T^{0}_{Q}(X,Y)\ \ (\text{ cf.
\eqref{e:u_Q} })\ $

  3. c)

$\ \ \ u_{Q}(X,Y)\ =\ 0\ \ (\text{ cf. \eqref{e:u_Q} }),\qquad L_{ij}(\nabla
h)=0.$

###### Proof.

Use (7.22) and (5.6) to get $2\nabla dh(X,Y)\ +\
2dh(\xi_{j})\,\omega_{j}(X,Y)\ +\ 2g(T^{0}_{Q}X,Y)\ =\ \nu g(X,Y).$
Decomposing in the $[-1]$ and $[3]$ components completes the proof of a) and
b), taking into account (7.23) and Lemma 7.12. The skew-symmetric part of
(7.23) gives $2u_{Q}\ +\ \sum_{(ijk)}L_{ij}(\nabla h)\omega_{k}\ =\ 0,$ where
the sum is over all even permutations of $(1,2,3)$. Hence, c) follows by
comparing the trace and trace-free parts of the last equality. ∎

##  8\. Quaternionic contact Yamabe problem

###  8.1. The Divergence Formula

Let $(M,\eta)$ be a quaternionic contact manifold with a fixed globally
defined contact form $\eta$. For a fixed $j\in\\{1,2,3\\}$ the form

(8.1) $Vol_{\eta}=\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge\omega_{j}^{2n}$

is a volume form. Note that $Vol_{\eta}$ is independent of $j$. We define the
(horizontal) divergence of a horizontal vector field/one-form
$\sigma\in\Lambda^{1}\,(H)$ by

(8.2) $\nabla^{*}\,\sigma\ =tr|_{H}\nabla\sigma=\
\sum_{a=1}^{4n}(\nabla_{e_{\alpha}}\sigma)(e_{\alpha}).$

Clearly the horizontal divergence does not depend on the basis and is an
$Sp(n)Sp(1)$-invariant. For any horizontal 1-form $\sigma\in\Lambda^{1}\,(H)$
we denote with $\sigma^{\\#}$ the corresponding horizontal vector field via
the horizontal metric defined with the equality $\sigma(X)=g(\sigma^{\\#},X)$.
It is justified to call the function $\nabla^{*}\,\sigma$ divergence of
$\sigma$ in view of the following Proposition.

######  Proposition 8.1.

Let $(M,\eta)$ be a quaternionic contact manifold of dimension (4n+3) and
$\eta\wedge\omega_{s}^{2n-1}\ \overset{def}{=}\
\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge\omega_{s}^{2n-1}.$ For any
horizontal 1-form $\sigma\in\Lambda^{1}\,(H)$ we have

$d(\sigma^{\\#}\lrcorner(Vol_{\eta}))\ =\
-(\nabla^{*}\sigma)\,\eta\wedge\omega^{2n}\ .$

Therefore, if $M$ is compact,

$\int_{M}(\nabla^{*}\sigma)\,\eta\wedge\omega^{2n}\ =\ 0.$

###### Proof.

We work in a qc-normal frame at a point $p\in M$ constructed in Lemma 4.5.
Since $\sigma$ is horizontal, we have
$\sigma^{\\#}=g(\sigma^{\\#},e_{a})e_{a}$. Therefore, we calculate

$\sigma^{\\#}\lrcorner(Vol_{\eta})=\sum_{a=1}^{4n}(-1)^{a}g(\sigma^{\\#},e_{a})\eta\wedge
e^{\\#}_{a_{1}}\wedge\dots\wedge\hat{e^{\\#}_{a}}\wedge\dots\wedge
e^{\\#}_{4n},$

where $\hat{e^{\\#}_{a}}$ means that the 1-form $e^{\\#}_{a}$ is missing in
the above wedge product. The exterior derivative of the above expression gives

$d(\sigma^{\\#}\lrcorner(Vol_{\eta}))=-\sum_{a=1}^{4n}e_{a}g(\sigma^{\\#},e_{a})Vol_{\eta}=-(\nabla^{*}\sigma)\,Vol_{\eta}.$

Indeed, since the Biquard connection preserves the metric, the middle term is
calculated as follows
$e_{a}g(X,e_{a})=g(\nabla_{e_{a}}X,e_{a})+g(X,\nabla_{a_{a}}e_{a})$ which
evaluated at the point $p$ gives

$e_{a}g(X,e_{a})_{|_{p}}=g(\nabla_{e_{a}}X,e_{a})_{|_{p}})=((\nabla_{e_{a}}\sigma)e_{a})_{|_{p}}.$

In order to obtain the last equality we also used the definition of the Reeb
vector fields, (2.5), and the following sequence of identities

$de^{\\#}_{b}(e_{b},e_{a})_{|_{p}}=e^{\\#}_{b}([e_{b},e_{a}])_{|_{p}}=e^{\\#}_{b}(\nabla_{e_{b}}e_{a}-\nabla_{e_{a}}e_{b}-T(e_{b},e_{a}))_{|_{p}}=0$

since $T(e_{b},a_{a})$ is a vertical vector field. This proves the first
formula. If the manifold is compact, then Stoke’s theorem completes the proof.
∎

We note that the integral formula of the above theorem was essentially proved
in [[W], Proposition 2.1].

###  8.2. Partial solutions of the QC-Yamabe problem

In this Section we shall present a partial solution of the Yamabe problem on
the quaternionic sphere. Equivalently, using the Cayley transform this
provides a partial solution of the Yamabe problem on the quaternionic
Heisenberg group. The extra assumption under which we classify the solutions
of the Yamabe equation consists of assuming that the ”new” quaternionic
structure has an integrable vertical space. The change of the vertical space
is given by (5.1). Of course, the standard quaternionic contact structure has
an integrable vertical distribution. A note about the Cayley transform is in
order. We shall define below the explicit Cayley transform for the considered
case, but one should keep in mind the more general setting of groups of
Heisenberg type [CDKR1]. In that respect, the solutions of the Yamabe equation
on the quaternionic Hesenberg group, which we describe, coincide with the
solutions on the groups of Heisenebrg type [GV1].

As in Section 5 we are considering a conformal transformation
$\tilde{\eta}=\frac{1}{2h}\eta$, where $\tilde{\eta}$ represents a fixed
quaternionic contact structure and $\eta$ is the ”new” structure conformal to
the original one. In fact, eventually, $\tilde{\eta}$ will stand for the
standard quaternionic contact structure on the quaternionic sphere. The Yamabe
problem in this case is to find all structures $\eta$, which are conformal to
$\tilde{\eta}$ and have constant scalar curvature equal to $16n(n+2)$, see
Corollary 4.14. The Yamabe equation is given by (5.10) and the problem is to
find all solutions of this equation.

######  Proposition 8.2.

Let $(M,\bar{\eta})$ be a compact qc-Einstein manifold of dimension $(4n+3)$.
Let $\bar{\eta}=\frac{1}{2h}\eta$ be a conformal deformation of the qc-
structure $\bar{\eta}$ on $M$. Suppose $\eta$ has constant scalar curvature.

  1. a)

If $n>1$, then any one of the following two conditions

    1. i)

the vertical space of $\eta$ is integrable;

    2. ii)

the QC structure $\eta$ is qc-pseudo Einstein;

implies that $\eta$ is a qc-Einstein structure.

  2. b)

If $n=1$ and the vertical space of $\eta$ is integrable than $\eta$ is a qc-
Einstein structure.

###### Proof.

The proof follows the steps of the solution of the Riemannian Yamabe problem
on the standard unit sphere, see [LP]. Theorem 1.3 tells us that
$\tilde{\eta}$ is a qc-Einstein structure. Theorem 3.12 and equations (5.9),
(5.7), and (5.8) imply

(8.3) $\displaystyle[Ric_{0}]_{[-1]}(X,Y)\ $ $\displaystyle=\
(2n+2)T^{0}(X,Y)\ =\ -(2n+2)h^{-1}[\nabla dh]_{[sym][-1]}(X,Y)$ (8.4)
$\displaystyle[Ric_{0}]_{[3]}(X,Y)\ $ $\displaystyle=\ 2(2n+5)U(X,Y)\ =\
-(2n+5)h^{-1}[\nabla dh-2h^{-1}dh\otimes dh]_{[3][0]}(X,Y).$

Furthermore, when the scalar curvature of $\eta$ is a constant then Theorem
4.8 gives

(8.5)
$\nabla^{*}T^{0}=(n+2)\mathbb{A},\qquad\nabla^{*}U=\frac{(1-n)}{2}\mathbb{A}.$

If $n>1$ and either the vertical space of $\eta$ is an integrable distribution
or $\eta$ is qc-pseudo Einstein $U=0$, then (8.5) shows that $\mathbb{A}=0$
and the divergences of $T^{0}$ and $U$ vanish $\nabla^{*}\,T^{0}\ =\ 0$ and
$\nabla^{*}\,U\ =\ 0$. The same conclusion can be reached in the case $n=1$
assuming the integrability of the vertical space (recall that always $U=0$
when $n=1$). We shall see that, in fact, $T^{0}$ and $U$ vanish, i.e., $\eta$
is also qc-Einstein. Consider first the $[-1]$ component. Taking norms,
multiplying by $h$ and integrating, the divergence formula gives

$\int_{M}h\ \lvert\,[Ric_{o}]_{[-1]}\ \rvert^{2}\,\eta\wedge\omega^{2n}\ =\
(2n+2)\ \int\langle[Ric_{o}]_{[-1]},\nabla
dh]\rangle\,\eta\wedge\omega^{2n}\\\ =(2n+2)\int_{M}\langle\nabla^{*}\
[Ric_{o}]_{[-1]},\nabla h]\rangle\,\eta\wedge\omega^{2n}\ =\ 0.\hskip
72.26999pt$

Thus, the $[-1]$ component of the qc-Einstein tensor vanishes
$\lvert\,[Ric_{o}]_{[-1]}\ \rvert\ =\ 0$. Define $h=\frac{1}{2u}$, inserting
(6.2) into (8.4) one gets

$[Ric_{0}]_{[3]}\ =\ 2(2n+5)U\ =\ -(2n+5)[\nabla du]_{[3][0]},$

from where, arguing as before we get $[Ric_{0}]_{[3]}\ =\ 0$. Theorem 1.3
completes the proof. ∎

######  Corollary 8.3.

Let $\bar{\eta}=\frac{1}{2h}\eta$ be a conformal deformation of a compact qc-
Einstein manifold of dimension $(4n+3)$ and suppose $\bar{\eta}$ has constant
qc-scalar curvature.

  1. i)

If $n>1$ and either the gradient $\nabla h$ or the gradient
$\nabla(\frac{1}{h})$ is a QC vector fields then $h$ is a constant.

  2. ii)

If $n=1$ and the gradient $\nabla(\frac{1}{h})$ is a QC vector fields then $h$
is a constant.

###### Proof.

Suppose $\nabla h$ is a QC-vector field. Corollary 7.15, b) yields $[\nabla
dh]_{[sym][-1]}=0$ since the torsion of Biquard connection vanishes due to
Proposition 4.2. Then Proposition 8.2 and a) in Corollary 7.15 imply that on
$H$ we have

$dh\otimes dh+d_{1}h\otimes d_{1}h+d_{2}h\otimes d_{2}h+d_{3}h\otimes
d_{3}h=\frac{|dh|^{2}}{n}g.$

If $n>1$ then $dh_{|H}=0$, which implies $dh=0$ using the bracket generating
condition.

Suppose $\nabla(\frac{1}{h})$ is a QC vector field. Then Proposition 8.2,
(6.2) combined with b) in Corollary 7.15 show that on $H$ we have

$3dh\otimes dh-d_{1}h\otimes d_{1}h-d_{2}h\otimes d_{2}h-d_{3}h\otimes
d_{3}h=0.$

Define $X=I_{1}X,Y=I_{1}Y$ etc. to get $dh\otimes dh=d_{1}h\otimes
d_{1}h=d_{2}h\otimes d_{2}h=d_{3}h\otimes d_{3}h$. Hence, $dh_{|H}=0$ since
dim $Ker\,dh\ =\ 4n-1$ and $dh=0$ as above. ∎

###  8.3. Proof of Theorem 1.2

###### Proof.

We start the proof with the observation that from Proposition 8.2 and
Corollary 6.26 the new structure $\eta$ is also qc-Einstein. Next we bring
into consideration the quaternionic Heisenberg group. Let us identify
$\boldsymbol{G\,(\mathbb{H})}$ with the boundary $\Sigma$ of a Siegel domain
in $\mathbb{H}^{n}\times\mathbb{H}$,

$\Sigma\ =\ \\{(q^{\prime},p^{\prime})\in\mathbb{H}^{n}\times\mathbb{H}\ :\
\Re{\ p^{\prime}}\ =\ \lvert q^{\prime}\rvert^{2}\\},$

by using the map $(q^{\prime},\omega^{\prime})\mapsto(q^{\prime},\lvert
q^{\prime}\rvert^{2}-\omega^{\prime})$. The standard contact form, written as
a purely imaginary quaternion valued form, is given by (cf. (5.17))
$2\tilde{\Theta}=(d\omega\ -\ q^{\prime}\cdot d\bar{q}^{\prime}\ +\
dq^{\prime}\,\cdot\bar{q}^{\prime}),$ where $\cdot$ denotes the quaternion
multiplication. Since $dp^{\prime}\ =\ q^{\prime}\cdot d\bar{q}^{\prime}\ +\
dq^{\prime}\,\cdot\bar{q}^{\prime}\ -\ d\omega^{\prime},$ under the
identification of $\boldsymbol{G\,(\mathbb{H})}$ with $\Sigma$ we have also
$2\tilde{\Theta}\ =\ -dp^{\prime}\ +\ 2dq^{\prime}\cdot\bar{q}^{\prime}.$
Taking into account that $\tilde{\Theta}$ is purely imaginary, the last
equation can be written also in the following form

$4\,\tilde{\Theta}\ =\ (d\bar{p}^{\prime}\ -\ dp^{\prime})\ +\
2dq^{\prime}\cdot\bar{q^{\prime}}\ -\ 2q^{\prime}\cdot d\bar{q}^{\prime}.$

Now, consider the Cayley transform as the map $\mathcal{C}:S\mapsto\Sigma$
from the sphere $S\ =\ \\{\lvert q\rvert^{2}+\lvert
p\rvert^{2}=1\\}\subset\mathbb{H}^{n}\times\mathbb{H}$ minus a point to the
Heisenberg group $\Sigma$, with $\mathcal{C}$ defined by

$(q^{\prime},p^{\prime})\ =\ \mathcal{C}\ \Big{(}(q,p)\Big{)},\qquad
q^{\prime}\ =\ (1+p)^{-1}\ q,\qquad p^{\prime}\ =\ (1+p)^{-1}\ (1-p)$

and with an inverse map $(q,p)\ =\
\mathcal{C}^{-1}\Big{(}(q^{\prime},p^{\prime})\Big{)}$ given by

$q\ =\ \ 2(1+p^{\prime})^{-1}\ q^{\prime},\qquad p\ =\ (1+p^{\prime})^{-1}\
(1-p^{\prime}).$

The Cayley transform maps $S$ minus a point to $\Sigma$ since

$\Re{\ p^{\prime}}\ =\ \Re{\frac{(1+\bar{p})(1-p)}{\lvert 1+p\,\rvert^{2}}}\
=\ \Re{\frac{1-\lvert p\rvert}{\lvert 1+p\,\rvert^{2}}}\ =\ \frac{\lvert
q\rvert^{2}}{\lvert 1+p\,\rvert^{2}}\ =\ \lvert q^{\prime}\rvert^{2}.$

Writing the Cayley transform in the form $(1+p)q^{\prime}\ =\ \
q,\quad(1+p)p^{\prime}\ =\ 1-p,$ gives

$dp\cdot q^{\prime}\ +\ (1+p)\cdot dq^{\prime}\ =\ dq,\hskip 36.135ptdp\cdot
p^{\prime}\ +\ (1+p)\cdot dp^{\prime}\ =\ -dp,$

from where we find

(8.8) $\displaystyle dp^{\prime}$ $\displaystyle=\ -2(1+p)^{-1}\cdot
dp\cdot(1+p)^{-1}$ $\displaystyle dq^{\prime}$ $\displaystyle=\
(1+p)^{-1}\cdot[dq\ -\ dp\cdot(1+p)^{-1}\cdot q].$

The Cayley transform is a conformal quaternionic contact diffeomorphism
between the quaternionic Heisenberg group with its standard quaternionic
contact structure $\tilde{\Theta}$ and the sphere minus a point with its
standard structure $\tilde{\eta}$, a fact which can be seen as follows.
Equations (8.8) imply the following identities

(8.15) $2\,\mathcal{C}^{*}\,\tilde{\Theta}\ =\ -(1+\bar{p})^{-1}\cdot
d\bar{p}\cdot(1+\bar{p})^{-1}\ +\ (1+p)^{-1}\cdot dp\cdot(1+p)^{-1}\\\ +\
(1+p)^{-1}\,[dq\ -\ dp\cdot(1+p)^{-1}\cdot
q]\cdot\bar{q}\cdot(1+\bar{p})^{-1}\\\ -\ (1+p)^{-1}\,q\cdot[d\bar{q}\ -\
\bar{q}\cdot(1+\bar{p})^{-1}\cdot d\bar{p}\,]\cdot(1+\bar{p})^{-1}\\\ =\
(1+p)^{-1}\,\Big{[}dp\cdot(1+p)^{-1}\cdot(1+\bar{p})\ -\ \lvert
q\rvert^{2}\,dp\cdot(1+p)^{-1}\Big{]}(1+\bar{p})^{-1}\\\ +\
(1+p)^{-1}\,\Big{[}-(1+p)\cdot(1+\bar{p})^{-1}\cdot d\bar{p}\ +\ \lvert
q\rvert^{2}\,(1+p)^{-1}d\bar{p}\,\Big{]}(1+\bar{p})^{-1}\\\
+(1+p)^{-1}\,\Big{[}dq\cdot\bar{q}\ -\ q\cdot
d\bar{q}\,\Big{]}(1+\bar{p})^{-1}\ =\ \frac{1}{\lvert
1+p\,\rvert^{2}}\,\lambda\,\tilde{\eta}\,\bar{\lambda},$

where $\lambda\ ={\lvert 1+p\,\rvert}\,{(1+p)^{-1}}$ is a unit quaternion and
$\tilde{\eta}$ is the standard contact form on the sphere,

(8.16) $\tilde{\eta}\ =\ dq\cdot\bar{q}\ +\ dp\cdot\bar{p}\ -\ q\cdot
d\bar{q}-\ p\cdot d\bar{p}.$

Since $\lvert 1+p\rvert\ =\ \frac{2}{\lvert 1+p^{\prime}\rvert}$ we have
$\lambda\ =\ \frac{1+p^{\prime}}{\lvert 1+p^{\prime}\,\rvert}$ equation (8.15)
can be put in the form

$\lambda\ \cdot(\mathcal{C}^{-1})^{*}\,\tilde{\eta}\ \cdot\bar{\lambda}\ =\
\frac{8}{\lvert 1+p^{\prime}\,\rvert^{2}}\,\tilde{\Theta}.$

We see that up to a constant multiplicative factor and a quaternionic contact
automorphism the forms $(\mathcal{C}^{-1})^{*}\tilde{\eta}$ and
$\tilde{\Theta}$ are conformal to each other. It follows that the same is true
for $(\mathcal{C}^{-1})^{*}\eta$ and $\tilde{\Theta}$.

In addition, $\tilde{\Theta}$ is qc-Einstein by definition, while $\eta$ and
hence also $(\mathcal{C}^{-1})^{*}\eta$ are qc-Einstein as we observed at the
beginning of the proof. Now we can apply Theorem 1.1 according to which up to
a multiplicative constant factor the forms
$(\mathcal{C}^{-1})^{*}\tilde{\eta}$ and $(\mathcal{C}^{-1})^{*}\eta$ are
related by a translation or dilation on the Heisenebrg group. Hence, we
conclude that up to a multiplicative constant, $\eta$ is obtained from
$\tilde{\eta}$ by a conformal quaternionic contact automorphism, see
Definition 7.6. ∎

Let us note that the Cayley transform defined in the setting of groups of
Heisenberg type is also a conformal transformation on $H$, see cf. [ACD, Lemma
2.5]. One can write the above transformation formula in this more general
setting.

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