# Invariant metrics for the quaternionic Hardy space

Nicola Arcozzi, Giulia Sarfatti Partially supported by the PRIN project Real
and Complex Manifolds of the Italian MIUR and by INDAM-GNAMPAPartially
supported by INDAM-GNSAGA and by the PRIN project Real and Complex Manifolds
of the Italian MIUR

###### Abstract

We find Riemannian metrics on the unit ball of the quaternions, which are
naturally associated with reproducing kernel Hilbert spaces. We study the
metric arising from the Hardy space in detail. We show that, in contrast to
the one-complex variable case, no Riemannian metric is invariant under regular
self-maps of the quaternionic ball.
Key words and phrases: Hardy space on the quaternionic ball; functions of a
quaternionic variable; invariant Riemannian metric.
Mathematics Subject Classification: 30G35, 46E22, 58B20.

Notation. The symbol $\mathbb{H}$ denotes the set of the quaternions
$q=x_{0}+x_{1}i+x_{2}j+x_{3}k=\operatorname{Re}(q)+\operatorname{Im}(q)$, with
$\operatorname{Re}(q)=x_{0}$ and $\operatorname{Im}(q)=x_{1}i+x_{2}j+x_{3}k$;
where the $x_{j}$’s are real numbers and the imaginary units $i,j,k$ are
subject to the rules $ij=k,\ jk=i,\ ki=j$ and $i^{2}=j^{2}=k^{2}=-1$. We
identify the quaternions $q$ whose imaginary part vanishes,
$\operatorname{Im}(q)=0$, with real numbers,
$\operatorname{Re}(q)\in\mathbb{R}$; and, similarly, we let
$\mathbb{I}=\mathbb{R}i+\mathbb{R}j+\mathbb{R}k$ be the set of the imaginary
quaternions. The norm $|q|\geq 0$ of $q$ is
$|q|=\sqrt{\sum_{l=0}^{3}x_{l}^{2}}=(q\overline{q})^{1/2}$, where
$\overline{q}=x_{0}-x_{1}i-x_{2}j-x_{3}k$ is the conjugate of $q$. The open
unit ball $\mathbb{B}$ in $\mathbb{H}$ contains the quaternions $q$ such that
$|q|<1$. The boundary of $\mathbb{B}$ in $\mathbb{H}$ is denoted by
$\partial\mathbb{B}$. By the symbol $\SS$ we denote the unit sphere of the
imaginary quaternions: $q\in\mathbb{I}$ belongs to $\SS$ if $|q|=1$. For $I$
in $\mathbb{S}$, the slice $L_{I}=L_{-I}$ in $\mathbb{H}$ contains all
quaternions having the form $q=x+yI$, with $x,y$ in $\mathbb{R}$. If $f$ is a
real differentiable function on a domain $\Omega\subseteq\mathbb{H}$, we
denote its real differential at a point $w\in\Omega$ by the symbol $f_{*}[w]$.

##  1 Introduction

Let $\mathbb{H}$ be the skew-field of the quaternions. The quaternionic Hardy
space $H^{2}(\mathbb{B})$ consists of the formal power series of the
quaternionic variable $q$,

$f(q)=\sum_{n=0}^{\infty}q^{n}a_{n},$

such that the sequence of quaternions $\\{a_{n}\\}$ satisfies

$\|f\|_{H^{2}(\mathbb{B})}:=\left(\sum_{n=0}^{\infty}\left|a_{n}\right|^{2}\right)^{1/2}<\infty.$
(1)

It is easily verified that the series converges to a function
$f:\mathbb{B}=\\{q\in\mathbb{H}:\ |q|<1\\}\to\mathbb{H}$. The function $f$ is
slice regular [13] in the sense of Gentili and Struppa, who developed a
version of complex function theory which holds in the quaternionic setting.
See the monograph [12] for a detailed account of the theory. The norm (1) can
be polarized to obtain an inner product with values in the quaternions,

$\left\langle\sum q^{n}a_{n},\sum
q^{n}b_{n}\right\rangle_{H^{2}(\mathbb{B})}:=\sum_{n=0}^{\infty}\overline{b_{n}}a_{n}.$

The space $H^{2}(\mathbb{B})$ is a reproducing kernel Hilbert space, in the
quaternionic sense: for $w$ in $\mathbb{B}$ and $f$ in $H^{2}(\mathbb{B})$ we
have

$f(w)=\left\langle f,k_{w}\right\rangle_{H^{2}(\mathbb{B})},\text{ where
}k_{w}(q)=k(w,q)=\sum_{n=0}^{\infty}q^{n}\overline{w}^{n}.$

There is a rich interplay between reproducing kernel Hilbert spaces and
distance functions. See [3] for an overview and several examples from one-
variable holomorphic function space theory. In [9] the connection between
metric theory and operator theory is analyzed at a very deep level, and the
case of the Hardy space is a model example of that. The seminal article [4] by
Aronszajn is still an excellent introduction to the theory of reproducing
kernel Hilbert spaces.

In this article we are mainly interested in studying metrics on $\mathbb{B}$
which are associated with the function space $H^{2}(\mathbb{B})$. We also
provide evidence that the metric properties of the space reflect the behavior
of functions in $H^{2}(\mathbb{B})$. The first metric we consider measures the
distance between projections of kernel functions in the unit sphere of the
Hilbert space $H^{2}(\mathbb{B})$:

$\delta(p,q):=\sqrt{1-\left|\left\langle\frac{k_{q}}{\|k_{q}\|_{H^{2}(\mathbb{B})}},\frac{k_{p}}{\|k_{p}\|_{H^{2}(\mathbb{B})}}\right\rangle_{H^{2}(\mathbb{B})}\right|^{2}}.$
(2)

In the holomorphic case of $H^{2}(\Delta)$ one obtains this way the pseudo-
hyperbolic metric
$\delta^{\prime}(z,w)=\left|\frac{z-w}{1-\overline{w}z}\right|$. A
calculation, see Proposition 4.2 below, gives a formally similar result in the
quaternionic case:

$\delta(z,w)=|(1-q\overline{w})^{-*}*(q-w)|_{|_{q=z}},$

for $z,w$ in $\mathbb{B}$. Here, the product $f(q)*g(q)$ and the
multiplicative inverse $f(q)^{-*}$ are not pointwise product and pointwise
inverse: they are $*$-product and $*$-inverse, which are defined so that the
usual convolution rule for coefficients of power series’ products holds. See
[12], and Section 2 where we summarize some background material on slice
regular functions.

The infinitesimal version of the pseudo-hyperbolic metric in the complex disc,
is the hyperbolic metric in the Riemann-Beltrami-Poincaré disc model:
$ds^{2}=\frac{|dz|^{2}}{(1-|z|^{2})^{2}}$. By infinitesimal version of a
distance $\delta_{0}$, we mean the length metric associated with $\delta_{0}$
(see e.g. [15]). The infinitesimal metric associated with $\delta$ is a
Riemannian metric $g$ on $\mathbb{B}$. In Theorem 3.2 a formula is produced,
which works for a wide class of reproducing kernel quaternionic Hilbert
spaces. Here is the special case of the quaternionic Hardy space.

######  Theorem 1.1.

(I)  The length metric associated with (2) is the Riemannian metric $g$
defined below.

For any $w\in\mathbb{B}$, let us identify the tangent space $T_{w}\mathbb{B}$
with $\mathbb{H}$. For any vector $d\in T_{w}\mathbb{B}$, where $w=x+yI_{w}$
lies in $L_{I_{w}}$, we decompose $d=d_{1}+d_{2}$ with $d_{1}$ in $L_{I_{w}}$
and $d_{2}$ in $L_{I_{w}}^{\perp}$, the orthogonal complement of $L_{I_{w}}$
with respect to the Euclidean metric in $\mathbb{H}$. The length of $d$ with
respect to $g$ is:

$|d|_{g(w)}^{2}=\frac{1}{(1-|w|^{2})^{2}}|d_{1}|^{2}+\frac{1}{|1-w^{2}|^{2}}|d_{2}|^{2}.$
(3)

(II) The isometry group of $g$ is the one generated by the following classes
of self-maps of $\mathbb{B}$:

  * (a)

regular Möbius transformations of the form

$q\mapsto
M_{\lambda}(q)=(1-q\lambda)^{-*}*(q-\lambda)=\frac{q-\lambda}{1-\lambda q},$

with $\lambda$ in $(-1,1)$;

  * (b)

isometries of the sphere of the imaginary units,

$q=x+yI\mapsto T_{A}(q)=x+yA(I),$

where $A:\SS\to\SS$ is an isometry of $\,\SS$;

  * (c)

the reflection in the imaginary hyperplane,

$q\mapsto R(q)=-\overline{q}.$

In the metric (3), the first, “large” summand is the hyperbolic metric on a
slice, while the second “small” summand is peculiarly quaternionic: it
measures the variation of a quaternionic Hardy function in the direction
orthogonal to the slices. Its small size reflects in quantitative, geometric
terms the fact that regular functions are affine in the $\SS$ variable, see
[12].

The special rôle of the real axis in the theory of slice regular functions is
here reflected in the fact that all isometries of the metric $g$ fix
$\mathbb{R}\cap\mathbb{B}$. In particular, contrary to the case of the complex
disc, the action is not transitive. Other, more precise, properties of the
metric will be stated and proved on route to the proof of Theorem 1.1. We will
study geodesics, geodesically complete submanifolds and other geometric
properties of the metric. For instance, we will prove that the radius of
injectivity is infinite at points of the real diameter of $\mathbb{B}$, and
finite elsewhere. The metric has neither positive, nor negative sectional
curvature.

The proof of Theorem 1.1 is split into two steps. In Theorem 3.2 we will
compute the Riemannian metric associated with rather general reproducing
kernel quaternionic Hilbert spaces $\mathcal{H}$; that is the length metric
associated with the distance function

$\delta_{\mathcal{H}}(w,z)=\sqrt{1-\left|\left\langle\frac{k_{w}}{\|k_{w}\|_{\mathcal{H}}},\frac{k_{z}}{\|k_{z}\|_{\mathcal{H}}}\right\rangle_{\mathcal{H}}\right|^{2}}.$

We restrict most of our analysis to spaces of functions defined on symmetric
slice domanis in $\mathbb{H}$, with a slice preserving reproducing kernel.
Examples are the Hardy space on $\mathbb{B}$ and on the right half-space
$\mathbb{H}^{+}=\\{q\in\mathbb{H}\,|\,\operatorname{Re}(q)>0\\}$ and the
Bergman space on $\mathbb{B}$. The classification of the isometries is carried
out in Section 4.

The Riemannian metric $g$ has a rather small group of isometries, compared
with the state of things in the unit disc of the complex plane, or even in the
unit ball in several complex variables, with the Bergman-Kobayashi metric. The
latter metrics have a transitive group of isometries and, more, the space is
isotropic; whereas all isometries of the former have to fix the real line. One
might expect that something better is possible. Unfortunately, there is no
Riemannian metric on $\mathbb{B}$ which is invariant under regular Möbius
functions and which is “democratic” with respect to the sphere of imaginary
units. If a geometric invariant for slice regular functions on the
quaternionic ball exists, it has to be other than a Riemannian metric.

######  Theorem 1.2.

There is no Riemannian metric $m$ on $\mathbb{B}$ having as isometries:

  * (i)

regular Möbius transformations $q\mapsto(1-q\overline{a})^{-*}*(q-a)u$, with
$a$ in $\mathbb{B}$ and $|u|=1$;

  * (ii)

maps of the form $q=x+yI\mapsto x+yA(I)$, with $A$ in $O(3)$, the orthogonal
group of $\mathbb{R}^{3}$.

The proof will be given in Subsection 4.3. We mention here that Bisi and
Gentili proved in [5] that the usual Poincaré metric on $\mathbb{B}$ is
invariant under classical (non-regular) Möbius maps.

A first relationship between the space $H^{2}(\mathbb{B})$ and the metric $g$
concerns the $H^{2}$ norm itself. Let $r\SS^{3}$ be the sphere of radius
$0<r<1$ in $\mathbb{B}$, with respect to the usual Euclidean metric;
containing quaternions $q=re^{tI}$, with $I$ in $\SS$ and $t$ in $[0,\pi]$.
The restriction of $g$ to $r\SS^{3}$ induces a volume form
$dVol_{r\mathbb{S}^{3}}$. Let $f$ be in $H^{2}(\mathbb{B})$. Then,

$\|f\|_{H^{2}(\mathbb{B})}^{2}=\lim_{r\to
1}(1-r^{2})\frac{1}{Vol_{r\mathbb{S}^{3}}(r\mathbb{S}^{3})}\int_{r\mathbb{S}^{3}}|f|^{2}dVol_{r\mathbb{S}^{3}},$

a relation similar to the definition of the Hardy norm in the unit disc by
means of the Poincaré metric.

In Section 5 we use the Caley map $C:q\mapsto(1-q)^{-1}(1+q)$ to write down
the metric $g$ in coordinates living in right half-space
$\mathbb{H}^{+}:=C(\mathbb{B})=\\{q\in\mathbb{H}\,|\,\operatorname{Re}(q)>0\\}$.
This makes it easier to prove a bilateral estimate for the distance function
associated with $g$, Theorem 5.2. As an application, in Theorem 5.4 we further
investigate the “rigidity” of the metric $g$, by showing that the only inner
functions which are Lipschitz continuous with respect to $g$ have to be be
slice preserving. In particular, they have to fix the real diameter of
$\mathbb{B}$. A function defined from $\Omega\subseteq\mathbb{H}$ to
$\mathbb{H}$ is slice preserving if it maps $L_{I}\cap\Omega$ to $L_{I}$ for
all $I$ in $\SS$.

We also consider in Subsection 5.1 four equivalent definitions of the Hardy
space $H^{2}(\mathbb{H}^{+})$ on $\mathbb{H}^{+}$. First, functions $f$ in
$H^{2}(\mathbb{H}^{+})$ might be characterized, pretty much as in the one-
dimensional complex case, as inverse Fourier transforms -in the quaternionic
sense- of functions $F:[0,\infty)\to\mathbb{H}$ with finite $L^{2}$-norm

$\|F\|_{L^{2}}=\left(\int_{0}^{\infty}|F(\zeta)|^{2}d\zeta\right)^{1/2}.$

Equivalently, $H^{2}(\mathbb{H}^{+})$ is the Hilbert space having reproducing
kernel $k_{w}(q)=(q+\overline{w})^{-*}$. This second viewpoint has the
advantage of relating $H^{2}(\mathbb{H}^{+})$ and $H^{2}(\mathbb{B})$. We show
in fact in Proposition 5.1 that the reproducing kernel for
$H^{2}(\mathbb{H}^{+})$ is a rescaling of the reproducing kernel for
$H^{2}(\mathbb{B})$:

$k_{H^{2}(\mathbb{B})}(C^{-1}(w),C^{-1}(z))=\frac{1}{2}(1+z)k_{H^{2}(\mathbb{H}^{+})}(w,z)(1+\overline{w}).$

Here we use the symbols $k_{H^{2}(\mathbb{B})}$ and
$k_{H^{2}(\mathbb{H}^{+})}$ for the reproducing kernels on $\mathbb{B}$ and
$\mathbb{H}^{+}$, respectively. Hence, third, the functions in
$H^{2}(\mathbb{H}^{+})$ might be defined as the rescaled versions of functions
in $H^{2}(\mathbb{B})$. Fourth, the norm of $f$ in $H^{2}(\mathbb{H}^{+})$ can
be computed as the limit of the integrals of $|f|^{2}$ on “horocycles” in
$H^{2}(\mathbb{H}^{+})$, when these are endowed with the natural volume form
induced by the metric $g$. The space $H^{2}(\mathbb{H}^{+})$ was defined in
[2], using a fifth definition, which is shown to give rise to the same
reproducing kernel. Our contribution here is mainly relating
$H^{2}(\mathbb{H}^{+})$ and the geometry of $\mathbb{H}^{+}$.

##  2 Preliminaries

We recall the definition of slice regularity, together with some basic results
that hold for this class of functions. We refer to the book [12] for all
details and proofs. Let $\mathbb{H}$ denote the four-dimensional (non-
commutative) real algebra of quaternions and let $\mathbb{S}$ denote the two-
dimensional sphere of imaginary units of $\mathbb{H}$,
$\mathbb{S}=\\{q\in\mathbb{H}\,|\,q^{2}=-1\\}$. One can “slice” the space
$\mathbb{H}$ in copies of the complex plane that intersect along the real
axis,

$\mathbb{H}=\bigcup_{I\in\mathbb{S}}(\mathbb{R}+\mathbb{R}I),\hskip
28.45274pt\mathbb{R}=\bigcap_{I\in\mathbb{S}}(\mathbb{R}+\mathbb{R}I),$

where $L_{I}:=\mathbb{R}+\mathbb{R}I\cong\mathbb{C}$, for any
$I\in\mathbb{S}$. Then, each element $q\in\mathbb{H}$ can be expressed as
$q=x+yI_{q}$, where $x,y$ are real (if $q\in\mathbb{R}$, then $y=0$) and
$I_{q}$ is an imaginary unit. Let $\Omega\subseteq\mathbb{H}$ be a subset of
$\mathbb{H}$. For any $I\in\mathbb{S}$, we will denote by $\Omega_{I}$ the
intersection $\Omega\cap L_{I}$. We can now recall the definition of slice
regular functions, in the sequel simply called regular functions.

######  Definition 2.1.

Let $\Omega$ be a domain (open connected subset) in $\mathbb{H}$. A function
$f:\Omega\to\mathbb{H}$ is said to be (slice) regular if for any
$I\in\mathbb{S}$ the restriction $f_{I}$ of $f$ to $\Omega_{I}$ has continuous
partial derivatives and it is such that

$\overline{\partial}_{I}f_{I}(x+yI)=\frac{1}{2}\left(\frac{\partial}{\partial
x}+I\frac{\partial}{\partial y}\right)f_{I}(x+yI)=0$

for all $x+yI\in\Omega_{I}$.

A wide class of examples of regular functions is given by power series with
quaternionic coefficients of the form $\sum^{\infty}_{n=0}q^{n}a_{n}$ which
converge in open balls centered at the origin.

######  Theorem 2.2.

A function $f$ is regular on $B(0,R)=\\{q\in\mathbb{H}\,|\,|q|<R\\}$ if and
only if $f$ has a power series expansion $f(q)=\sum^{\infty}_{n=0}q^{n}a_{n}$
converging in $B(0,R)$.

For regular functions, it is possible to define an appropriate notion of
derivative:

######  Definition 2.3.

Let $f$ be a regular function on a domain $\Omega\subseteq\mathbb{H}$. The
slice (or Cullen) derivative of $f$ is the regular function defined as

$\partial_{c}f(x+yI)=\frac{1}{2}\left(\frac{\partial}{\partial
x}-I\frac{\partial}{\partial y}\right)f_{I}(x+yI).$

We will consider domains in certain restricted classes.

######  Definition 2.4.

Let $\Omega\subseteq\mathbb{H}$ be a domain.

  1. 1.

$\Omega$ is called a slice domain if it intersects the real axis and if, for
any $I\in\mathbb{S}$, $\Omega_{I}$ is a domain in $L_{I}$.

  2. 2.

$\Omega$ is called a symmetric domain if for any point $x+yI\in\Omega$, with
$x,y\in\mathbb{R}$ and $I\in\mathbb{S}$, the entire two-sphere $x+y\mathbb{S}$
is contained in $\Omega$.

The ball $\mathbb{B}$ and the right half-space $\mathbb{H}^{+}=\\{q=x+Iy:\
I\in\SS,\ x>0,\ y\in\mathbb{R}\\}$ are symmetric slice domains.

Slice regular functions defined on symmetric slice domains have a peculiar
property.

######  Theorem 2.5 (Representation Formula).

Let $f$ be a regular function on a symmetric slice domain $\Omega$ and let
$x+y\mathbb{S}\subset\Omega$. Then, for any $I,J\in\mathbb{S}$,

$f(x+yJ)=\frac{1}{2}[f(x+yI)+f(x-yI)]+J\frac{I}{2}[f(x-yI)-f(x+yI)].$

In particular, there exist $b,c\in\mathbb{H}$ such that $f(x+yJ)=b+Jc$ for any
$J\in\mathbb{S}$.

When restricted to a sphere of the form $x+y\mathbb{S}$, a regular function is
actually affine in the variable $q$. This nice geometric property leads to the
following definition

######  Definition 2.6.

Let $f$ be a regular function on a symmetric slice domain $\Omega$. The
spherical derivative of $f$ is defined as

$\partial_{s}f(q)=(q-\overline{q})^{-1}\left(f(q)-f(\overline{q})\right).$

A basic result that establishes a relation between regular functions and
holomorphic functions of one complex variable is the following.

######  Lemma 2.7 (Splitting Lemma).

Let $f$ be a regular function on a slice domain $\Omega\subseteq\mathbb{H}$.
Then for any $I\in\mathbb{S}$ and for any $J\in\mathbb{S}$, $J\perp I$ there
exist two holomorphic functions $F,G:\Omega_{I}\to L_{I}$ such that

$f(x+yI)=F(x+yI)+G(x+yI)J$

for any $x+yI\in\Omega_{I}$.

In general, the pointwise product of functions does not preserve slice
regularity. It is possible to introduce a new multiplication operation, which,
in the special case of power series, can be defined as follows.

######  Definition 2.8.

Let $f(q)=\sum_{n=0}^{\infty}q^{n}a_{n},$ and
$g(q)=\sum_{n=0}^{\infty}q^{n}b_{n}$ be regular functions on $B(0,R)$. Their
regular product (or $*$-product) is

$f*g(q)=\sum_{n\geq 0}q^{n}\sum_{k=0}^{n}a_{k}b_{n-k},$

regular on $B(0,R)$ as well.

The $*$-product is related to the standard pointwise product by the following
formula.

######  Proposition 2.9.

Let $f,g$ be regular functions on a symmetric slice domain $\Omega$. Then

$f*g(q)=\left\\{\begin{array}[]{l r}0&\text{if $f(q)=0$}\\\
f(q)g(f(q)^{-1}qf(q))&\text{if $f(q)\neq 0$}\end{array}\right.$

The reciprocal $f^{-*}$ of a regular function $f$ with respect to the
$*$-product can be defined.

######  Definition 2.10.

Let $f(q)=\sum_{n=0}^{\infty}q^{n}a_{n}$ be a regular function on $B(0,R)$,
$f\not\equiv 0$. Its regular reciprocal is

$f^{-*}(q)=\frac{1}{f*f^{c}(q)}f^{c}(q),$

where $f^{c}(q)=\sum_{n=0}^{\infty}q^{n}\overline{a}_{n}$. The function
$f^{-*}$ is regular on $B(0,R)\setminus\\{q\in B(0,R)\ |\ f*f^{c}(q)=0\\}$ and
$f*f^{-*}=1$ there.

For example, in the case of the reproducing kernel for the quaternionic Hardy
space $H^{2}(\mathbb{B})$, we have

######  Remark 2.11.

The reproducing kernel for $H^{2}(\mathbb{B})$ is

$k_{w}(q)=\sum_{n=0}^{\infty}q^{n}\overline{w}^{n}=(1-q\overline{w})^{-*}.$

Then we have a natural definition of _regular quotients_ of regular functions,
which satisfy

######  Proposition 2.12.

Let $f$ and $g$ be regular functions on a symmetric slice domain $\Omega$ and
denote by $Z=\\{q\in\Omega\ |\ f*f^{c}(q)=0\\}$. If $T_{f}:\Omega\setminus
Z\rightarrow\Omega\setminus$ is defined as

$T_{f}(q)=f^{c}(q)^{-1}qf^{c}(q),$

then

$f^{-*}*g(q)=f(T_{f}(q))^{-1}g(T_{f}(q))\quad\text{for every}\quad
q\in\Omega\setminus Z_{f^{s}}.$

Important examples of regular quotients that will appear in the sequel are the
regular Möbius transformations, of the form

$M_{a}(q)=(1-q\overline{a})^{-*}*(q-a),$

where $a\in\mathbb{B}$, which are regular self-maps of the quaternionic unit
ball $\mathbb{B}$. After multiplication on the right by unit-norm quaternions,
they are the only self-maps of $\mathbb{B}$ which are regular, with regular
inverse. They were introduced by Stoppato in [18]. See also [12].

##  3 Metrics associated with quaternionic reproducing kernel Hilbert spaces

Let $\Omega\subseteq\mathbb{H}$ be a symmetric slice domain and let
$\mathcal{H}$ be a reproducing kernel Hilbert space of regular functions on
$\Omega$. For the definition and all basic results concerning quaternionic
Hilbert spaces see, e.g., [14] and references therein. For the properties we
are interested in, the same results hold in quaternion valued Hilbert spaces
and complex valued Hilbert spaces, and the proofs are very similar. It is
possible to define a metric $\delta_{\mathcal{H}}$ on $\Omega$ in terms of the
distance between projections of kernel functions in the unit sphere of the
Hilbert space $\mathcal{H}$. Namely, if $k(w,q)=k_{w}(q)$ denotes the
reproducing kernel of $\mathcal{H}$, then
$\delta_{\mathcal{H}}:\Omega\times\Omega\to\mathbb{R}^{+}$ can be defined as

$\delta_{\mathcal{H}}(w,z)=\sqrt{1-\left|\left\langle\frac{k_{w}}{\|k_{w}\|_{\mathcal{H}}},\frac{k_{z}}{\|k_{z}\|_{\mathcal{H}}}\right\rangle_{\mathcal{H}}\right|^{2}}.$
(4)

######  Proposition 3.1.

Let $\Omega$ be a symmetric slice domain and let $\mathcal{H}$ be a
reproducing kernel Hilbert space of regular functions on $\Omega$. Let
$w\in\Omega\cap L_{I_{w}}$ and let $d\in\mathbb{H}$ be such that
$w+d\in\Omega$. Consider the decomposition $d=d_{1}+d_{2}$, where $d_{1}\in
L_{I_{w}}$ and $d_{2}\in L_{I_{w}}^{\perp}$. Then

$\delta^{2}_{\mathcal{H}}(w,w+d)=\frac{\|k_{w}\|_{\mathcal{H}}^{2}\left\|\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\|_{\mathcal{H}}^{2}-\left|\left\langle
k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right|^{2}}{\|k_{w}\|_{\mathcal{H}}^{4}}+O(|d|^{2}).$

###### Proof.

Recalling the definition (4) of $\delta_{\mathcal{H}}$, we get

$\delta^{2}_{\mathcal{H}}(w,w+d)=\frac{\|k_{w}\|^{2}_{\mathcal{H}}\|k_{w+d}\|^{2}_{\mathcal{H}}-\left|\left\langle
k_{w},k_{w+d}\right\rangle_{\mathcal{H}}\right|^{2}}{\|k_{w}\|^{2}_{\mathcal{H}}\|k_{w+d}\|^{2}_{\mathcal{H}}}.$
(5)

We want to have a better description of the numerator of (5). Using the
properties of the kernel functions and the fact that regular functions are
real analytic functions of $4$ real variables, we can write

$k_{w+d}(q)-k_{w}(q)=\overline{k_{q}(w+d)-k_{q}(w)}=\overline{(k_{q})_{*}[w](d)}+O(|d|^{2})$

where $(k_{q})_{*}[w](d)$ denotes the real differential of $k_{q}$ at the
point $w$, applied to the vector $d$. We identify here the tangent space
$T_{w}\Omega$ with $\mathbb{H}$. Thanks to the decomposition properties of the
real differential of regular functions in terms of slice and spherical
derivatives, see Remark 8.15 in [12], we have

$k_{w+d}(q)-k_{w}(q)=\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}+O(|d|^{2}),$

hence,

$\|k_{w+d}\|_{\mathcal{H}}^{2}=\|k_{w}\|_{\mathcal{H}}^{2}+\left\|\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\|_{\mathcal{H}}^{2}+2\operatorname{Re}\left\langle
k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}+O(|d|^{2})$

and

$\displaystyle\left|\left\langle
k_{w},k_{w+d}\right\rangle_{\mathcal{H}}\right|^{2}=\left|\|k_{w}\|_{\mathcal{H}}^{2}+\left\langle
k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right|^{2}+O(|d|^{2})$
$\displaystyle=\|k_{w}\|_{\mathcal{H}}^{4}+\left|\left\langle
k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right|^{2}$
$\displaystyle\hskip
170.71652pt+2\|k_{w}\|_{\mathcal{H}}^{2}\operatorname{Re}\left\langle
k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}+O(|d|^{2}).$

Therefore

$\delta^{2}_{\mathcal{H}}(w,w+d)=\frac{\|k_{w}\|_{\mathcal{H}}^{2}\left\|\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\|_{\mathcal{H}}^{2}-\left|\left\langle
k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right|^{2}}{\|k_{w}\|_{\mathcal{H}}^{4}}+O(|d|^{2}).$

∎

Proposition 3.1 reflects what happens in the complex case, see [16]. In fact
the functions $\overline{\partial_{c}k_{q}(w)}$ and
$\overline{\partial_{s}k_{q}(w)}$ are regular with respect to the variable
$q$, and they reproduce respectively the slice and the spherical derivative of
any regular function $f:\Omega\to\mathbb{H}$. In fact, for any
$w\in\Omega_{I_{w}}$, if $h\in L_{I_{w}}$, we can write

$\displaystyle\partial_{c}f(w)=\lim_{h\to 0,\,h\in
L_{I_{w}}}h^{-1}(f(w+h)-f(w))=\lim_{h\to 0,\,h\in
L_{I_{w}}}h^{-1}\left(\left\langle
f,k_{w+h}\right\rangle_{\mathcal{H}}-\left\langle
f,k_{w}\right\rangle_{\mathcal{H}}\right)$ $\displaystyle=\lim_{h\to 0,\,h\in
L_{I_{w}}}h^{-1}\left\langle
f,\overline{k_{q}(w+h)-k_{q}(w)}\right\rangle_{\mathcal{H}}=\lim_{h\to
0,\,h\in L_{I_{w}}}h^{-1}\left\langle
f,\overline{h\partial_{c}k_{q}(w)}\right\rangle_{\mathcal{H}}=\left\langle
f,\overline{\partial_{c}k_{q}(w)}\right\rangle_{\mathcal{H}}$

and

$\displaystyle\partial_{s}f(w)$
$\displaystyle=(w-\overline{w})^{-1}(f(w)-f(\overline{w}))=(w-\overline{w})^{-1}\left(\langle
f,k_{w}\rangle_{\mathcal{H}}-\langle
f,k_{\overline{w}}\rangle_{\mathcal{H}}\right)$
$\displaystyle=(w-\overline{w})^{-1}\left\langle
f,k_{w}-k_{\overline{w}}\right\rangle_{\mathcal{H}}=(w-\overline{w})^{-1}\left\langle
f,\overline{(w-\overline{w})\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}=\left\langle
f,\overline{\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}.$

Proposition 3.1 allows us to define a Riemannian metric $g_{\mathcal{H}}$ on
the symmetric slice domain $\Omega$. For each point $w\in\Omega$, let us
identify the tangent space $T_{w}\Omega$ with
$\mathbb{H}=L_{I_{w}}+L_{I_{w}}^{\perp}$. Then the length of a tangent vector
$d=d_{1}+d_{2}\in L_{I_{w}}+L_{I_{w}}^{\perp}$ is

$|d|^{2}_{{g_{\mathcal{H}}}(w)}=\frac{\|k_{w}\|_{\mathcal{H}}^{2}\left\|\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\|_{\mathcal{H}}^{2}-\left|\left\langle
k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right|^{2}}{\|k_{w}\|_{\mathcal{H}}^{4}}.$
(6)

######  Theorem 3.2.

Let $\Omega$ be a symmetric slice domain and let $\mathcal{H}$ be a
reproducing kernel Hilbert space of regular functions on $\Omega$. Suppose
that $k$ is slice preserving: for any $w\in\Omega$ the kernel function $k_{w}$
preserves the slice $L_{I_{w}}$ identified by $w$. Then the length of a
tangent vector $d=d_{1}+d_{2}\in L_{I_{w}}+L_{I_{w}}^{\perp}\cong T_{w}\Omega$
with respect to the Riemannian metric $g_{\mathcal{H}}$ associated with
$\mathcal{H}$ is given by

$|d|^{2}_{{g_{\mathcal{H}}}(w)}=\frac{\left(\|k_{w}\|_{\mathcal{H}}^{2}\left\|\overline{\partial_{c}k_{q}(w)}\right\|_{\mathcal{H}}^{2}-\left|\overline{\partial_{c}k_{w}(w)}\right|^{2}\right)}{\|k_{w}\|_{\mathcal{H}}^{4}}|d_{1}|^{2}+\frac{\left(\|k_{w}\|_{\mathcal{H}}^{2}\left\|\overline{\partial_{s}k_{q}(w)}\right\|_{\mathcal{H}}^{2}-\left|\overline{\partial_{s}k_{w}(w)}\right|^{2}\right)}{\|k_{w}\|_{\mathcal{H}}^{4}}|d_{2}|^{2}.$

###### Proof.

We begin by working out the numerator in equation (6). We have:

$\displaystyle\left\|\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\|_{\mathcal{H}}^{2}=\left\langle\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}$
$\displaystyle=\left\|\overline{\partial_{c}k_{q}(w)}\right\|_{\mathcal{H}}^{2}|d_{1}|^{2}+\left\|\overline{\partial_{s}k_{q}(w)}\right\|_{\mathcal{H}}^{2}|d_{2}|^{2}+2\operatorname{Re}\left\langle\overline{d_{1}\partial_{c}k_{q}(w)},\overline{d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}$

and

$\displaystyle\left|\left\langle
k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right|^{2}=\left|\overline{d_{1}\partial_{c}k_{w}(w)+d_{2}\partial_{s}k_{w}(w)}\right|^{2}$
$\displaystyle=\left|\overline{\partial_{c}k_{w}(w)}\right|^{2}|d_{1}|^{2}+\left|\overline{\partial_{s}k_{w}(w)}\right|^{2}|d_{2}|^{2}+2\operatorname{Re}\left(\overline{d_{1}\partial_{c}k_{w}(w)}d_{2}\partial_{s}k_{w}(w)\right).$

Hence we are left to prove that both

$\operatorname{Re}\left\langle\overline{d_{1}\partial_{c}k_{q}(w)},\overline{d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}=\operatorname{Re}\left(d_{2}\left\langle\overline{\partial_{c}k_{q}(w)},\overline{\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\overline{d_{1}}\right)$

and

$\operatorname{Re}\left(\overline{d_{1}\partial_{c}k_{w}(w)}d_{2}\partial_{s}k_{w}(w)\right)=\operatorname{Re}\left(d_{2}\partial_{s}k_{w}(w)\overline{\partial_{c}k_{w}(w)}\,\overline{d_{1}}\right)$

equal zero. Now notice that if $k_{w}$ maps $\Omega_{I_{w}}$ to $L_{I_{w}}$,
the same holds true for both $\partial_{c}k_{w}$ and $\partial_{s}k_{w}$. The
fact that $d_{1}\in L_{I_{w}}$ and $d_{2}\in L_{I_{w}}^{\perp}$ leads us to
conclude.

∎

The hypothesis about kernel functions required in Theorem 3.2 is satisfied by
the quaternionic analogues of Hardy and Bergman spaces; see [1, 8].

##  4 Invariant metrics associated with the Hardy space $H^{2}(\mathbb{B})$

In this section, we turn our attention to the special example of the Hardy
space $H^{2}(\mathbb{B})$. We will study the corresponding Riemannian metric
$g:=g_{H^{2}(\mathbb{B})}$. Recalling that for any $w$ the kernel function
$k_{w}(q)=\sum^{\infty}_{n=0}q^{n}\overline{w}^{n}$ preserves the slice
$L_{I_{w}}$, we can directly apply Theorem 3.2 to find the expression of $g$,
thus proving the first part of Theorem 1.1.

######  Proposition 4.1.

For any $w\in\mathbb{B}$, let us identify the tangent space $T_{w}\mathbb{B}$
with $\mathbb{H}$. For any vector $d\in T_{w}\mathbb{B}$, if $w$ lies in
$L_{I_{w}}$ and we decompose $d=d_{1}+d_{2}$ with $d_{1}$ in $L_{I_{w}}$ and
$d_{2}$ in $L_{I_{w}}^{\perp}$, then the length of $d$ with respect to $g$ is
given by

$|d|_{g(w)}^{2}=\frac{1}{(1-|w|^{2})^{2}}|d_{1}|^{2}+\frac{1}{|1-w^{2}|^{2}}|d_{2}|^{2}.$
(7)

###### Proof.

The following equalities can, by their nature, be reduced to simple
calculations in the complex plane:

$\|k_{w}\|_{H^{2}(\mathbb{B})}^{2}=\frac{1}{1-|w|^{2}},\quad\left|\overline{\partial_{c}k_{w}(w)}\right|^{2}=\frac{|w|^{2}}{(1-|w|^{2})^{4}},\quad\left|\overline{\partial_{s}k_{w}(w)}\right|^{2}=\frac{|w|^{2}}{(1-|w|^{2})^{2}|1-w^{2}|^{2}},$
$\left\|\overline{\partial_{c}k_{q}(w)}\right\|_{H^{2}(\mathbb{B})}^{2}=\Big{\|}\sum_{n\geq
0}n^{2}q^{n}\overline{w}^{n-1}\Big{\|}_{H^{2}(\mathbb{B})}^{2}=\frac{1+|w|^{2}}{(1-|w|^{2})^{3}},$
$\left\|\overline{\partial_{s}k_{q}(w)}\right\|_{H^{2}(\mathbb{B})}^{2}=\frac{1}{|w-\overline{w}|^{2}}\left(\frac{2}{1-|w|^{2}}-\frac{1}{1-w^{2}}-\frac{1}{1-\overline{w}^{2}}\right).$

A direct application of Theorem 3.2, then, yields that, with respect to
coordinates $(d_{1},d_{2})\in(L_{I_{w}},L_{I_{w}}^{\perp})$,

$|d|^{2}_{g(w)}=\frac{1}{(1-|w|^{2})^{2}}|d_{1}|^{2}+\frac{1}{|1-w^{2}|^{2}}|d_{2}|^{2}.$

∎

The volume form $dVol_{g}$ associated with the metric $g$ at any point
$w=x_{0}+x_{1}i+x_{2}j+x_{3}k\in\mathbb{B}$ is then

$dVol_{g}(w)=\frac{dVol_{Euc}(w)}{(1-|w|^{2})^{2}|1-w^{2}|^{2}},$

where $dVol_{Euc}(w)=dx_{0}dx_{1}dx_{2}dx_{3}$ is the usual Euclidean volume
element.

######  Proposition 4.2.

Let $\delta:=\delta_{H^{2}(\mathbb{B})}$ be defined as in (4). For any
$w,z\in\mathbb{B}$, $\delta(z,w)$ coincides both with the value at $z$ of the
regular Möbius transformation $M_{w}$ associated with $w$ and with the vaule
at $w$ of the regular Möbius transformation $M_{z}$ associated with $z$,
namely

$\delta(w,z)=\left|(1-q\overline{z})^{-*}*(q-z)\right|_{|_{q=w}}=\left|(1-q\overline{w})^{-*}*(q-w)\right|_{|_{q=z}}.$

###### Proof.

Let $w,z$ be two points in $\mathbb{B}$. By Proposition 2.12,

$|\langle
k_{w},k_{z}\rangle_{H^{2}(\mathbb{B})}|=|k_{w}(z)|=|(1-q\overline{w})^{-*}|_{|_{q=z}}=|1-\hat{z}\overline{w}|^{-1}$

where $\hat{z}=(1-zw)^{-1}z(1-zw)$, which implies

$\left|\left\langle\frac{k_{w}}{\|k_{w}\|_{H^{2}(\mathbb{B})}},\frac{k_{z}}{\|k_{z}\|_{H^{2}(\mathbb{B})}}\right\rangle_{H^{2}(\mathbb{B})}\right|^{2}=|1-\hat{z}\overline{w}|^{-2}\left(1-|w|^{2}\right)\left(1-|z|^{2}\right).$

Thus, since $|\hat{z}|=|z|$, we get

$\displaystyle\delta^{2}(w,z)$
$\displaystyle=1-\left|\left\langle\frac{k_{w}}{\|k_{w}\|_{H^{2}(\mathbb{B})}},\frac{k_{z}}{\|k_{z}\|_{H^{2}(\mathbb{B})}}\right\rangle_{H^{2}(\mathbb{B})}\right|^{2}$
$\displaystyle=\left|1-\hat{z}\overline{w}\right|^{-2}\left(\left|1-\hat{z}\overline{w}\right|^{2}-\left(1-|w|^{2}\right)\left(1-|z|^{2}\right)\right)$
$\displaystyle=\left|1-\hat{z}\overline{w}\right|^{-2}\left(\left(1-\hat{z}\overline{w}\right)\left(1-w\overline{\hat{z}}\right)-\left(1-w\overline{w}\right)\left(1-\hat{z}\overline{\hat{z}}\right)\right)=\left|1-\hat{z}\overline{w}\right|^{-2}\left(\hat{z}-w\right)\left(\overline{\hat{z}}-\overline{w}\right)$
$\displaystyle=\left|1-\hat{z}\overline{w}\right|^{-2}\left|\hat{z}-w\right|^{2}=\left|\left(1-\hat{z}\overline{w}\right)^{-1}\left(\hat{z}-w\right)\right|^{2}=\left|\left(1-q\overline{w}\right)^{-*}*(q-w)\right|_{|_{q=z}}^{2}$

where the last equality follows from Proposition 2.12.∎

The previous relation between the metric $\delta$ (which is the finite version
of the metric $g$) and regular Möbius transformations, is not unexpected. In
fact, as studied in [6], the real differential $(M_{w})_{*}$ of the regular
Möbius map $M_{w}$ associated with a point $w\in\mathbb{B}_{I_{w}}$ acts on
$L_{I_{w}}$ by right multiplication by $(1-|w|^{2})^{-1}$ and on
$L_{I_{w}}^{\perp}$ by right multiplication by $(1-\overline{w}^{2})^{-1}$.
Looking at equation (7), we see that the coefficients of the metric $g$ at the
point $w$ with respect to coordinates $(L_{I_{w}},L_{I_{w}}^{\perp})$ coincide
in modulus with the components of $(M_{w})_{*}$. Moreover, the fact that
$g(w)$ measures vectors in $L_{I_{w}}$ by multiplying their Euclidean length
by $\frac{1}{1-|w|^{2}}$ means that the restriction of $g$ to a slice $L_{I}$
is the classical Poincaré metric in the unit disc $\mathbb{B}_{I}$.

Using spherical coordinates, $\mathbb{B}=\\{re^{tI}\ |\ r\in[0,1),\
t\in[0,\pi],\ I\in\mathbb{S}\\},$ if $q=re^{tI}$ and we decompose the lenght
element $dq=dq_{1}+dq_{2}\in L_{I_{w}}+L_{I_{w}}^{\perp}$, then, since $dI$ is
orthogonal to $I$ (because $I$ is unitary) we have
$|d_{1}|^{2}=dr^{2}+r^{2}dt^{2}$ and $|d_{2}|^{2}=r^{2}\sin^{2}t|dI|^{2}$
where $|dI|$ denotes the usual two-dimensional sphere round metric on
$\mathbb{S}\cong\mathbb{S}^{2}$. Therefore we get the expression of the metric
tensor $ds^{2}_{g}$ associated with $g$ in spherical coordinates:

$ds_{g}^{2}=\frac{dr^{2}+r^{2}dt^{2}}{(1-r^{2})^{2}}+\frac{r^{2}\sin^{2}t|dI|^{2}}{(1-r^{2})^{2}+4r^{2}\sin^{2}t}.$
(8)

That is, $g$ is a warped product of the hyperbolic metric $g_{hyp}$ on the
complex unit disc with the standard round metric $g_{\mathbb{S}}$ on the two-
dimensional sphere [17].

###  4.1 Isometries and geodesics of $(\mathbb{B},g)$

From the expression (7) of $g$, it is clear that three families of functions
act isometrically on $(\mathbb{B},g)$:

  * (a)

regular Möbius transformation of the form

$q\mapsto
M_{\lambda}(q)=(1-q\lambda)^{-*}*(q-\lambda)=\frac{q-\lambda}{1-q\lambda},$

with $\lambda$ in $(-1,1)$;

  * (b)

isometries of the sphere of imaginary units, which in polar coordinates $r\geq
0,t\in[0,\pi],I\in\mathbb{S}$ read as

$q=re^{tI}\mapsto T_{A}(q)=re^{tA(I)},$

where $A:\SS\to\SS$ is an isometry of $\SS$;

  * (c)

the reflection in the imaginary hyperplane,

$q\mapsto R(q)=-\overline{q}.$

Our goal is to prove the following classification result, thus proving the
second part of Theorem 1.1.

######  Theorem 4.3.

The group $\Gamma$ of isometries of $(\mathbb{B},g)$ is generated by maps of
type $(a)$, $(b)$ and $(c)$.

The proof requires a few steps. To begin with, we identify three classes of
totally geodesic submanifolds of $\mathbb{B}$, each one related to a class of
isometries.

The first family is the one related to isometries of type $(a)$.

######  Lemma 4.4.

For any $I\in\mathbb{S}$, the two-dimensional submanifold of $\mathbb{B}$

$\mathbb{B}_{I}=\mathbb{B}\cap
L_{I}=\\{re^{tI}\in\mathbb{B}\,|\,r\in[0,1),t\in[0,2\pi]\\}$

is totally geodesic. In particular, for any $I\in\mathbb{S}$, $\mathbb{B}_{I}$
is an hyperbolic disc.

###### Proof.

Fix $I\in\mathbb{S}$ and let $g^{I}_{hyp}$ be the restriction of the metric
$g$ to $\mathbb{B}_{I}$, which is just the classical hyperbolic metric in the
unit disc. We will show that each geodesic of $(\mathbb{B}_{I},g_{hyp}^{I})$
is still a geodesic of $(\mathbb{B},g)$. Pick two points $w,z$ in
$\mathbb{B}_{I}$ and let $\gamma$ be the (hyperbolic) geodesic in
$\mathbb{B}_{I}$ joining $w$ with $z$, and
$\alpha(\tau)=r(\tau)\left(\cos(t(\tau))+\sin(t(\tau))I(\tau)\right)$ be a
parametrized curve which joins $w=\alpha(\tau_{0})$ with $z=\alpha(\tau_{1})$.
If $\pi_{I}\left(\alpha\right)$ denotes the piecewise regular curve obtained
by projecting $\alpha$ on $\mathbb{B}_{I}$,

$\pi_{I}\left(\alpha\right)(\tau)=r(\tau)\left(\cos(t(\tau))+\sin(t(\tau))I\right),$

since $|dI|$ is orthogonal to $\mathbb{B}_{I}$, we conclude

$\operatorname{length}(\alpha)\geq\operatorname{length}(\pi_{I}\left(\alpha\right))\geq\operatorname{length}(\gamma).$

∎

The second family of totally geodesic submanifolds is related to isometries of
type $(b)$. For any $I\in\mathbb{S}$, we denote by $\mathcal{C}(I)$ the great
circle obtained intersecting $\mathbb{S}$ with the plane $L_{I}^{\perp}$.

######  Lemma 4.5.

For any $J\in\mathbb{S}$, the three-dimensional submanifold of $\mathbb{B}$

$\mathbb{B}(\mathcal{C}(J))=\\{re^{tI}\in\mathbb{B}\,|\,r\in[0,1),t\in[0,\pi],I\in\mathcal{C}(J)\\}$

is totally geodesic.

###### Proof.

We will prove the statement by showing that the imaginary units identified by
all points lying on a same geodesic always belong to the same great circle of
$\mathbb{S}$. More precisely, let
$\gamma(\tau)=r(\tau)\left(\cos(t(\tau))+\sin(t(\tau))I(\tau)\right)$ be a
parametrized geodesic of $(\mathbb{B},g)$ such that

$\left\\{\begin{array}[]{l}\gamma(\tau_{0})=x_{0}+y_{0}I_{0}\\\
\gamma^{\prime}(\tau_{0})=v_{0}+w_{0}J_{0}\end{array}\right.$

We want to show that, for any $\tau$, the imaginary unit $I(\tau)$ of
$\gamma(\tau)$ belongs to the great circle of $\SS$ identified by $I_{0}$ and
$J_{0}$, namely that, for any $\tau$,
$I(\tau)\in\mathcal{C}:=\mathcal{C}\left(I_{0}\times J_{0}\right).$ Let
$\psi:\mathbb{S}\to\mathbb{S}$ be the reflection of $\mathbb{S}$ with respect
to $\mathcal{C}$. Then the curve
$\tilde{\gamma}(\tau)=r(\tau)\left(\cos(t(\tau))+\sin(t(\tau))\psi(I(\tau))\right)$
is a geodesic of $(\mathbb{B},g)$ such that

$\left\\{\begin{array}[]{l}\tilde{\gamma}(\tau_{0})=\gamma(\tau_{0})\\\
\tilde{\gamma}^{\prime}(\tau_{0})=\gamma^{\prime}(\tau_{0})\end{array}\right.$

since $\psi$ fixes $I_{0}$ and $J_{0}$. By the uniqueness of geodesics with
assigned initial conditions, we get that $\tilde{\gamma}(\tau)=\gamma(\tau)$
and hence that $\psi$ fixes $I(\tau)$ for any $\tau$. Therefore we conclude
that $I(\tau)\in\mathcal{C}$ for any $\tau\in I$. ∎

The third totally geodesic submanifold is the one related to the last class of
isometries, type $(c)$.

######  Lemma 4.6.

The three-dimensional submanifold of $\mathbb{B}$

$\mathbb{B}\left(\pi/2\right)=\\{re^{tI}\in\mathbb{B}\,|\,r\in[0,1),t=\pi/2,I\in\mathbb{S}\\}=\\{rI\,|\,r\in[0,1),I\in\mathbb{S}\\}$

is totally geodesic.

###### Proof.

The statement can be proven following the line of the proof of Lemma 4.5. The
ingredients are the fact that the map $R:\mathbb{B}\to\mathbb{B}$,
$q\mapsto-\overline{q}$ is an isometry which fixes (punctually)
$\mathbb{B}(\pi/2)$, and the uniquness of geodesics with assigned initial
conditions. ∎

Considering the intersection of totally geodesic submanifolds of type
$\mathbb{B}(C(J))$ with $\mathbb{B}(\pi/2)$ allows us to identify another
family of totally geodesic submanifolds of $\mathbb{B}$.

######  Corollary 4.7.

Let $\mathcal{C}(J)$ be a great circle in $\mathbb{S}$. Then the two-
dimensional submanifold
$D\left(\pi/2,\mathcal{C}(J)\right)\subset\mathbb{B}(\pi/2)$, defined as

$D\left(\pi/2,\mathcal{C}(J)\right)=\\{re^{tI}\in\mathbb{B}\,|\,r\in[0,1),t=\pi/2,I\in\mathcal{C}(J)\\}=\\{rI\in\mathbb{B}\,|\,r\in[0,1),I\in\mathcal{C}(J)\\},$

is totally geodesic.

######  Remark 4.8.

Notice that for the two-dimensional submanifold
$D\left({\pi}/2,\mathcal{C}(J)\right)$ the following orthogonality relation
holds:

$D\left({\pi}/2,\mathcal{C}(J)\right)\cap\mathbb{B}_{J}=\\{0\\}\text{ and
}T_{0}D\left({\pi}/2,\mathcal{C}(J)\right)=T_{0}\mathbb{B}_{J}^{\perp}.$

Moreover, applying Möbius maps of the form $M_{\lambda}$ to
$D\left({\pi}/2,\mathcal{C}(J)\right)$, we can extend the orthogonality
relation from the origin to all points in $\mathbb{B}\cap\mathbb{R}$. In this
way we obtain a family of totally geodesic submanifolds

$D\left(t,\mathcal{C}(J)\right)=M_{\lambda(t)}\left(D\left({\pi}/2,\mathcal{C}(J)\right)\right)$

that, for $t\in[0,\pi]$ and $J\in\mathbb{S}/\\{\pm 1\\}$, defines a foliation
of the manifold $\mathbb{B}$.

In order to have some understanding of the (global) behavior of the metric
$g$, let us investigate some metric properties of the discs of the type
$D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$. Since the imaginary units taken
into account belong to $\mathcal{C}(J)\cong\mathbb{S}^{1}$, we can change
coordinates, setting $I=e^{i\theta}$ and $|dI|=d\theta$, so that the metric
$g$, on $D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$, reduces to

$ds_{D}^{2}=\frac{dr^{2}}{(1-r^{2})^{2}}+\frac{r^{2}d\theta^{2}}{(1+r^{2})^{2}}.$

It is actually convenient to parametrize
$D\left(\frac{\pi}{2},\mathcal{C}(J)\right)\subset\mathbb{I}\cong\mathbb{R}^{3}$
as a surface of revolution of the form
$(\Phi(\rho),\Psi(\rho)\cos\theta,\Psi(\rho)\sin\theta)$, where $\rho$ is the
arc length of the generating curve. Setting

$\rho=\rho(r)=\frac{1}{2}\log\frac{1+r}{1-r},$

we get

$\frac{dr^{2}}{(1-r^{2})^{2}}=d\rho^{2}\quad\text{and}\quad\frac{r^{2}}{(1+r^{2})^{2}}=\frac{1}{4}\tanh^{2}(2\rho)$

and hence, in coordinates $(\rho,\theta)$, we get that the metric is expressed
as

$ds_{D}^{2}=d\rho^{2}+\frac{1}{4}\tanh^{2}(2\rho)d\theta^{2}=d\rho^{2}+\Psi^{2}(\rho)d\theta^{2}.$
(9)

######  Remark 4.9.

The Gaussian curvature $K$ of the two-dimensional submanifold
$D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$ is positive. In fact, see e.g.
[11], with respect to coordinates $(\rho,\theta)$ it can be computed as

$K=\frac{-\Psi^{\prime\prime}(\rho)}{\Psi(\rho)}$

which is a non-negative quantity since $\Psi(\rho)=\frac{1}{2}\tanh(2\rho)\geq
0$ and $\Psi^{\prime\prime}(\rho)\leq 0$. This in particular implies that the
sectional curvature of $(\mathbb{B},g)$ is positive on all sections
$D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$, while it is negative on all
slices $\mathbb{B}_{I}$.

It is possible to study geodesics of
$D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$ by means of the Euler-Lagrange
equations

$\left\\{\begin{array}[]{l}\frac{\partial}{\partial\theta}L=\frac{d}{dt}\frac{\partial}{\partial\dot{\theta}}L\\\
\frac{\partial}{\partial\rho}L=\frac{d}{dt}\frac{\partial}{\partial\dot{\rho}}L\end{array}\right.$

associated with the Lagrangian

$L(\rho,\theta,\dot{\rho},\dot{\theta},\tau)=\frac{1}{2}\left(\dot{\rho}^{2}+\frac{\tanh^{2}(2\rho)}{4}\dot{\theta}^{2}\right),$

namely

$\left\\{\begin{array}[]{l}0=\frac{d}{dt}\left(\frac{\tanh^{2}(2\rho)}{4}\dot{\theta}\right)\\\
\tanh(2\rho)\frac{1-\tanh^{2}(2\rho)}{4}\dot{\theta}^{2}=\ddot{\rho}.\end{array}\right.$
(10)

The first equation in (10) yields

$\frac{\tanh^{2}(2\rho)}{4}\dot{\theta}=A,$

for some constant $A$. If $A=0$, we get $\dot{\theta}=0$ and hence the second
equation in (10) implies that $\ddot{\rho}=0$. If otherwise $A\neq 0$ we get
$\dot{\theta}=\frac{4A}{\tanh^{2}(2\rho)}$ which implies
$|\dot{\theta}|>4|A|$. Therefore all generating curves, with $\dot{\theta}=0$,
are geodesics of $D$. Which is not surprising since they correspond to radii
$\gamma(r)=re^{\frac{\pi}{2}I}$ for $I\in\mathcal{C}(J)$. The other important
fact that arises is that for any point $q\in
D\left(\frac{\pi}{2},\mathcal{C}(J)\right)\setminus\\{0\\}$ any geodesic
corresponding to $A\neq 0$ intersects in finite time the “radial” geodesic
through $q$. This leads to the following result.

######  Lemma 4.10.

Let $J\in\mathbb{S}$. For any $q\in
D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$ such that $q\neq 0$, the
injectivity radius at $q$ is finite. On the other hand, the injectivity radius
at $q=0$ is infinite.

This important metric property of the point $q=0$ is useful to classify the
isometries of $(\mathbb{B},g)$. First of all it tells us that isometries map
the real diameter of $\mathbb{B}$ to itself.

######  Lemma 4.11.

Let $\Gamma$ be the group of isometries of $(\mathbb{B},g)$. Then, for any
$\phi\in\Gamma$, $\phi(\mathbb{B}\cap\mathbb{R})=\mathbb{B}\cap\mathbb{R}$.

###### Proof.

Consider first $q=0$. Since the injectivity radius at $q=0$ is infinite, then,
for any $\phi\in\Gamma$, the injectivity radius at $\phi(0)$ is infinite as
well. By post-composing $\phi$ with a regular Möbius transformation of type
$(a)$ $M_{\lambda}$ we can map $0$ to
$D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$ (for some $J\in\mathbb{S}$) and
hence Lemma 4.10 yields that $M_{\lambda}(\phi(0))=0$. Since $M_{\lambda}$
preserves the real diameter of $\mathbb{B}$, we get that
$\phi(0)\in\mathbb{R}$. To conclude, notice that we can map each point of
$\mathbb{B}\cap\mathbb{R}$ to $0$ by means of a regular Möbius map of type
$(a)$. ∎

We can finally prove the Classification Theorem for isometries of
$(\mathbb{B},g)$.

###### Proof of Theorem 4.3.

Let $\Phi\in\Gamma$ be an isometry of $(\mathbb{B},g)$. Up to composition with
a regular Möbius transformation of type $(a)$ and with the map
$R:q\mapsto-\overline{q}$, we can suppose that $\Phi(0)=0$ and that, by Lemma
4.11, $\Phi(\mathbb{B}\cap\mathbb{R}^{+})=\mathbb{B}\cap\mathbb{R}^{+}$.

We now show that $\Phi$ fixes $\mathbb{B}(\pi/2)$. Set
$\tilde{B}(\pi/2)=\Phi(\mathbb{B}(\pi/2))$. Since $\Phi$ is an isometry, Lemma
4.6 implies that $\tilde{B}(\pi/2)$ is a totally geodesic submanifold of
$\mathbb{B}$. Moreover, since $\Phi(0)=0$, since the geodesics starting at $0$
lie on slices, and since, by Lemma 4.4, the slices carry the usual hyperbolic-
Poincaré metric: we have that $\Phi$ maps radii
$\gamma_{I}(r)=re^{\frac{\pi}{2}I}$ to radii of the form
$\Phi(\gamma_{I}(r))=re^{\theta(I)\psi(I)}$ with $\theta(I)\in[0,\pi]$, and
$\psi(I)\in\mathbb{S}$. Let us show that $\theta$ is actually constant on
$\mathbb{S}$.
If $d_{g}$ denotes the distance function on $\mathbb{B}$ associated with $g$,
recalling equation (8), on the one hand we have

$\displaystyle
d_{g}\left(\Phi(\gamma_{I_{1}}(r)),\Phi(\gamma_{I_{2}}(r))\right)$
$\displaystyle=$ $\displaystyle
d_{g}\left(\gamma_{I_{1}}(r),\gamma_{I_{2}}(r)\right)=d_{g}\left(re^{\frac{\pi}{2}I_{1}},re^{\frac{\pi}{2}I_{2}}\right)$
(11) $\displaystyle\leq$
$\displaystyle\frac{r}{1+r^{2}}d_{\mathbb{S}}(I_{1},I_{2})\stackrel{{\scriptstyle
r\to 1}}{{\longrightarrow}}\frac{1}{2}d_{\mathbb{S}}(I_{1},I_{2}),$ (12)

where $d_{\mathbb{S}}$ denotes the usual spherical distance on the unit sphere
$\mathbb{S}$. In particular we deduce that

$d_{g}\left(\Phi(\gamma_{I_{1}}(r)),\Phi(\gamma_{I_{2}}(r))\right)$

is bounded as a function of $r$. On the other hand, if
$\alpha(\tau)=r(\tau)e^{t(\tau)I(\tau)}$ is a parametrized geodesic joining
$\alpha(\tau_{1})=\Phi(\gamma_{I_{1}}(r))$ and
$\alpha(\tau_{2})=\Phi(\gamma_{I_{2}}(r))$, we have

$\displaystyle
d_{g}\left(\Phi(\gamma_{I_{1}}(r)),\Phi(\gamma_{I_{2}}(r))\right)=d_{g}\left(re^{\theta({I_{1}})\psi({I_{1}})},re^{\theta({I_{2}})\psi({I_{2}})}\right)=\operatorname{length}\left(\alpha([\tau_{1},\tau_{2}])\right)$
$\displaystyle=\int_{\tau_{1}}^{\tau_{2}}\sqrt{\frac{r^{\prime}(\tau)^{2}+r^{2}t^{\prime}(\tau)^{2}}{1-r(\tau)^{2}}+\frac{r(\tau)^{2}\sin^{2}(t(\tau))I^{\prime}(\tau)^{2}}{(1-r(\tau)^{2})^{2}+4r(\tau)^{2}\sin^{2}(t(\tau))}}d\tau$
$\displaystyle\geq\int_{\tau_{1}}^{\tau_{2}}\sqrt{\frac{r^{\prime}(\tau)^{2}+r(\tau)^{2}t^{\prime}(\tau)^{2}}{1-r(\tau)^{2}}}d\tau\geq
d_{hyp}(re^{\theta({I_{1}})I},re^{\theta({I_{2}})I})$

where $I$ is any fixed imaginary unit and $d_{hyp}$ the hyperbolic distance
associated to the restriction $g^{I}_{hyp}$ of the metric $g$ to
$\mathbb{B}\cap L_{I}$. If, by contradiction,
$\theta(I_{1})\neq\theta(I_{2})$, then the distance
$d_{hyp}(re^{\theta({I_{1}})I},re^{\theta({I_{2}})I})$ tends to infinity as
$r$ goes to $1$, contradicting (11). Then,
$\theta(I_{1})=t(\tau_{1})=t(\tau_{2})=\theta(I_{2})$: $\theta$ is constant on
$\mathbb{S}$.

Therefore we have that $\tilde{B}(\pi/2)$ is ruled by radii of the form
$\tilde{\gamma}(r)=re^{t_{0}I}$ for some constant $t_{0}$. If $t_{0}=\pi/2$,
then we are done. Suppose then $t_{0}\neq\pi/2$. Since $\tilde{B}(\pi/2)$ and
$\mathbb{B}(\pi/2)$ intersect at $0$ and they are three-dimensional
submanifolds in $\mathbb{H}$, the intersection $V$ of their respective tangent
spaces at $0$ must have dimension $2$ or $3$. Let $v$ be a vector in $V$ and
let $r\mapsto re^{Jt}$ be the reparametrized geodesic with initial velocity
$v$. The geodesic lies on both $\tilde{B}(\pi/2)$ and $\mathbb{B}(\pi/2)$,
hence $t_{0}=t=\pi/2$. (A different proof consists in showing that, if
$t_{0}\neq\pi/2$, then $\tilde{B}(\pi/2)$ is not smooth at the origin).

The next step is to show that the restriction of $\Phi$ to $\mathbb{B}(\pi/2)$
is an isometry $T_{A}$ of type $(b)$ for some isometry $A$ of the sphere
$\mathbb{S}$. We have that

$\displaystyle
d_{g}\left(\Phi(\gamma_{I_{1}}(r)),\Phi(\gamma_{I_{2}}(r))\right)=d_{g}\left(re^{\frac{\pi}{2}\psi({I_{1}})},re^{\frac{\pi}{2}\psi({I_{2}})}\right).$
(13)

We now prove an improvement of (11).

######  Lemma 4.12.

$\lim_{r\to
1}d_{g}\left(re^{\frac{\pi}{2}I_{1}},re^{\frac{\pi}{2}I_{2}}\right)=\frac{1}{2}d_{\mathbb{S}}(I_{1},I_{2}).$

###### Proof of the lemma..

Only the case $I_{1}\neq I_{2}$ is interesting. Let
$D\left(\pi/2,\mathcal{C}(J)\right)$ be the two-dimensional manifold
introduced in Lemma 4.7 which contains the reparametrized geodesics $r\mapsto
re^{\frac{\pi}{2}I_{j}}$, $j=1,2$. The metric $g$ restricted to the totally
geodesic surface $D\left(\pi/2,\mathcal{C}(J)\right)$ was discussed earlier in
this subsection, where we gave it the expression (9). Since
$\Psi^{\prime}(\rho)=1/\cosh(2\rho)\leq 1$, the surface can be isometrically
imbedded as a surface $S$ in $\mathbb{R}^{3}$, with parametric equations
$(u(s,\theta),v(s,\theta),z(s,\theta))=\chi(s,\theta)$, where:

$\begin{cases}u=p(s)\cos(\theta);\crcr v=p(s)\sin(\theta);\crcr
z=s.\end{cases}$

Here $s\geq 0$, $\theta\in[-\pi,\pi]$ and $p:[0,+\infty)\to[0,1/2)$ is a
smooth, increasing function such that $p(0)=0$ and
$\lim_{s\to\infty}p(s)=1/2$. The relationship between $p$ and $\Psi$ is the
following: if $\int_{0}^{s}\sqrt{p^{\prime}(\sigma)^{2}+1}d\sigma=\rho,$ then
$p(s)=\psi(\rho)$. Now, $r=constant\to 1$ corresponds to
$s=constant\to\infty$, and the choice of $I_{1}$ and $I_{2}$ corresponds to a
choice of $\theta_{1}$ and $\theta_{2}$. Let $k$ be the metric on the surface.
It is elementary that

$\lim_{s\to\infty}d_{k}(\chi(s,\theta_{1}),\chi(s,\theta_{2}))=\frac{1}{2}d_{\mathbb{S}^{1}}(\theta_{1},\theta_{2})$

is one-half the distance between $\theta_{1}$ and $\theta_{2}$ on the unit
circle, which is the same as one-half the distance between $I_{1}$ and $I_{2}$
in $\SS$. ∎

Equations (11) and (13) together with Lemma 4.12 imply that that
$\psi:\mathbb{S}\to\mathbb{S}$ is an isometry of the sphere $\mathbb{S}$, i.e.
$\Phi|_{\mathbb{B}(\pi/2)}=T_{\psi}|_{\mathbb{B}(\pi/2)}$. In conclusion,
$T_{\psi}^{-1}\circ\Phi$ is an isometry that fixes $\mathbb{R}\cap\mathbb{B}$
and $\mathbb{B}(\pi/2)$ and hence its (real) differential at the origin
$(T_{\psi}^{-1}\circ\Phi)_{*}[0]:T_{0}\mathbb{B}\to T_{0}\mathbb{B}$ is the
identity map. Therefore $T_{\psi}^{-1}\circ\Phi$ is the identity map as well
and the theorem is proved. ∎

###  4.2 Relation with the space $H^{2}(\mathbb{B})$

If we restrict the metric $g$ to a three-dimensional sphere $r\mathbb{S}^{3}$
of radius $r$, in spherical coordinates we get

$ds^{2}_{r\mathbb{S}^{3}}=\frac{r^{2}}{(1-r^{2})^{2}}dt^{2}+\frac{r^{2}\sin^{2}t}{(1-r^{2})^{2}+4r^{2}\sin^{2}(t)}|dI|^{2}$

whose corresponding volume form is

$dVol_{r\mathbb{S}^{3}}(re^{tI})=\frac{r^{3}\sin^{2}t}{(1-r^{2})((1-r^{2})^{2}+4r^{2}\sin^{2}(t))}dtdA_{\mathbb{S}}(I)$

where $dA_{\mathbb{S}}$ denotes the area element of the two-dimensional sphere
$\mathbb{S}$. This volume form (after a normalization) induces a volume form
on the boundary $\mathbb{S}^{3}$ of the unit ball: if
$u=e^{sJ}\in\mathbb{S}^{3}$, we have

$\displaystyle dVol_{\mathbb{S}^{3}}(u)$ $\displaystyle:=\lim_{r\to
1^{-}}(1-r^{2})dVol_{r\mathbb{S}^{3}}(ru)=\lim_{r\to
1^{-}}\frac{(1-r^{2})r^{3}\sin^{2}s}{(1-r^{2})((1-r^{2})^{2}+4r^{2}\sin^{2}(s))}dtdA_{\mathbb{S}}(I)$
$\displaystyle=\frac{1}{4}dtdA_{\mathbb{S}}(I).$

Notice that $dVol_{\mathbb{S}^{3}}$ is the product of the usual spherical
metric on the two-dimensional sphere $\mathbb{S}$ with the metric $dt$ on
circles $\mathbb{S}^{3}_{I}$ which appears in the definition of Hardy spaces
given in [10]. Moreover in [10] it is proven that any $f\in H^{2}(\mathbb{B})$
has radial limit along almost any radius and hence, denoting (with a slight
abuse of notation) the radial limit by $f$ itself, we have

$\displaystyle\int_{\mathbb{S}^{3}}|f(u)|^{2}dVol_{\mathbb{S}^{3}}(u)$
$\displaystyle=\frac{1}{4}\int_{\mathbb{S}}\int_{0}^{\pi}|f(e^{tI})|^{2}dtdA_{\mathbb{S}}(I)=\frac{1}{8}\int_{\mathbb{S}}\int_{0}^{2\pi}|f(e^{tI})|^{2}dtdA_{\mathbb{S}}(I)$
$\displaystyle=\frac{1}{8}\int_{\mathbb{S}}||f||^{2}_{H^{2}(\mathbb{B})}dA_{\mathbb{S}}(I)=\frac{\pi}{2}||f||^{2}_{H^{2}(\mathbb{B})}.$

###  4.3 Proof of Theorem 1.2

We begin by showing that $(\mathbb{B},m)$ has constant negative curvature.
Heuristically, a Riemannian metric $m$ satisfying the assumption of the
theorem has an isometry group with dimension

$dim(\mathbb{B})+dim(\SS^{3})+dim(O(3))=4+3+3=10,$

which is maximal for a four-dimensional Riemannian manifold. Hence,
$(\mathbb{B},m)$ has constant curvature.

More precisely, we show that the isometry group $\mathcal{I}$ acts
transitively on orthonormal frames, a property which is known to imply
constant sectional curvature. Given points $a,b$ in $\mathbb{B}$ and
orthonormal frames $\\{e_{l}(a):\ l=0,1,2,3\\}$ and $\\{e_{l}(b):\
l=0,1,2,3\\}$ in $T_{a}\mathbb{B}$ and $T_{b}\mathbb{B}$, respectively, we
find an isometry $\varphi$ in $\mathcal{I}$ such that its (real) differential
$\varphi_{*}$ satisfies: $\varphi_{*}[a]e_{l}(a)=e_{l}(b)$. We in fact exhibit
$\varphi$ mapping $a$ to $0$ and such that $\varphi_{*}$ maps the chosen
orthonormal frame in $T_{a}\mathbb{B}$ to a fixed orthonormal basis
$e_{0}(0),e_{1}(0),e_{2}(0),e_{3}(0)$ of $T_{0}\mathbb{B}$, where $e_{0}(0)$
is the vector tangent to the positive real half-axis. The isometry
$M_{a}(q)=(1-q\overline{a})^{-*}*(q-a)$ maps $a$ to $0$, hence $(M_{a})_{*}$
sends $e_{l}(a)$ to a orthonormal frame in $T_{0}\mathbb{B}$ $e^{\prime}_{l}$
for $l=0,\dots,3$. For a suitable choice of $u$ with $|u|=1$, the isometry
$q\mapsto q\cdot u$ has differential mapping $e^{\prime}_{0}$ to $e_{0}(0)$,
and $e^{\prime}_{l}(0)$ to $e^{\prime\prime}_{l}(0)$ ($j=1,2,3$). The
isometries $q=x+yI\mapsto T_{A}(q)=x+yA(I)$, $A$ being a fixed element of
$O(3)$, all have differentials fixing $e_{0}(0)$. We can find one mapping
$e^{\prime\prime}_{l}(0)$ to $e_{l}(0)$ for $l=1,2,3$. The composition of
these three isometries is the desired isometry $\varphi$.

For $I\in\SS$, consider the subgroup $\mathcal{I}_{I}$ of $\mathcal{I}$ of the
isometries fixing the slice $\mathbb{B}_{I}$; which consists of the regular
Möbius maps $M_{a}$, with $a$ in $\mathbb{B}_{I}$, and of the maps $q\mapsto
q\cdot e^{tI}$. If $\chi_{I}:x+yI\mapsto x+yi$ is the natural bijection from
$\mathbb{B}_{I}$ to the unit disc in the complex plane,
$\chi_{I}\mathcal{I}_{I}\chi_{I}^{-1}$ identifies $\mathcal{I}_{I}$ with the
usual Poincaré group in the complex disc. Hence, the restriction of $m$ to
$\mathbb{B}_{I}$ is (isometric to) a constant multiple of the Poincaré metric,
which has constant negative curvature.

The hyperbolic metric $m$ is realized by the standard Poincaré model on the
ball $\mathbb{B}$. The metric $m$ restricted to $\mathbb{B}_{I}$ is realized
as $|d|^{2}_{m(w)}=\lambda^{2}\frac{|d|^{2}}{(1-|w|^{2})^{2}}$ (for $w$ in
$\mathbb{B}_{I}$ and $d$ in $T_{w}(\mathbb{B}_{I})$), with $\lambda$ which is
independent of $I$, since different slices intersect along the real diameter
of $\mathbb{B}$. We might set $\lambda=1$. Each slice $\mathbb{B}_{I}$ is
totally geodesic, since it is the set of the points fixed by an isometry of
the type $x+yJ\mapsto x+yB(J)$, where $B$ is an element of $O(3)$ fixing $\pm
I$ and no other element of $\SS$.

Then, the radii $r\mapsto ru=\gamma_{u}(r)$ (with $u$ fixed in $\mathbb{H}$,
$|u|=1$) are (reparametrizations of) geodesics of $(\mathbb{B},m)$, not just
of its restriction to a slice, and the distance function on each of them is
obtained by integrating $dr/(1-r^{2})$. The three-dimensional spheres
$r\mathbb{S}^{3}=\\{q:\ |q|=r\\}$ are then metric spheres centered at $0$ for
the metric $m$. By Gauss Lemma, they are orthogonal to the curves
$\gamma_{u}$. Fix $r$ in $(0,1)$. An argument similar to the one above shows
that the subgroup $\mathcal{I}_{0}$ of the isometries fixing $0$ acts
transitively on the bundle of the orthogonal frames at points of
$r\mathbb{S}^{3}$, hence $r\mathbb{S}^{3}$ is isometric to the usual three-
spheres with a multiple of the spherical metric. Since $L_{I}\cap
r\mathbb{S}^{3}$ is isometric to a metric one sphere in the Poincaré model of
the hyperbolic metric (in dimension two), for each $I$ in $\SS$,
$r\mathbb{S}^{3}$ is similarly isometric to a metric three-sphere in the
Poincaré model of the hyperbolic metric (in dimension four). But we said that
$r\mathbb{S}^{3}$ and $\gamma_{u}$ are orthogonal in their point of
intersection; they are both isometric to the corresponding objects in the
Poincaré model; the sum of their tangent spaces is the whole tangent space:
this shows that the metric $m$ coincides in fact with the Poincaré metric.

Concluding, we have shown that the hyperbolic Poincaré metric is invariant
under regular Möbius maps, but this contradicts a result of Bisi and Stoppato
[6], Remark 5.

##  5 Metric in the right half space $\mathbb{H}^{+}$

Consider the right half space
$\mathbb{H}^{+}=\\{q\in\mathbb{H}\,|\,\operatorname{Re}(q)>0\\}$. The Cayley
map $C:\mathbb{B}\to\mathbb{H}^{+}$,

$C(q)=(1-q)^{-1}(1+q),$

is a regular bijection from the quaternionic unit ball onto the quaternionic
right half space with regular inverse the function
$C^{-1}:\mathbb{H}^{+}\to\mathbb{B}$,

$C^{-1}(q)=(1+q)^{-1}(q-1).$

The aim of this section is to study the image $(\mathbb{H}^{+},h)$ of
$(\mathbb{B},g)$ under the map $C$, where $h$ is the pullback of the metric
$g$ by the map $C^{-1}$. In the introduction we labeled $g$ and $h$ by the
same letter, since $C$ is by definition an isometry from $(\mathbb{B},g)$ to
$(\mathbb{H}^{+},h)$. Let $u\in\mathbb{H}^{+}$ and let $v=v_{1}+v_{2}$ be a
tangent vector in $T_{u}\mathbb{H}^{+}\cong L_{I_{u}}+L_{I_{u}}^{\perp}$. The
length of $v$ with respect to $h$ is

$|v|_{h(u)}=\left|(C^{-1})_{*}[u](v)\right|_{g(C^{-1}(u))}$

where $(C^{-1})_{*}[u]$ is the real differential of $C^{-1}$ at the point
$u\in\mathbb{H}^{+}$. Recalling the decomposition of the real differential of
a regular function in terms of its slice and spherical derivatives, if
$v=v_{1}+v_{2}\in L_{I_{u}}+L_{I_{u}}^{\perp}$ we can write

$(C^{-1})_{*}[u](v)=v_{1}\partial_{c}C^{-1}(u)+v_{2}\partial_{s}C^{-1}(u)=v_{1}\frac{2}{(1+u)^{2}}+v_{2}\frac{2}{|1+u|^{2}}.$

Hence $(C^{-1})_{*}[u]$ preserves the decomposition
$T_{u}\mathbb{H}^{+}=L_{I_{u}}+L_{I_{u}}^{\perp}$ and we get

$\displaystyle|v|^{2}_{h(u)}$
$\displaystyle=\frac{1}{(1-|C^{-1}(u)|^{2})^{2}}\frac{4}{|1+u|^{4}}|v_{1}|^{2}+\frac{1}{|1-C^{-1}(u)^{2}|^{2}}\frac{4}{|1+u|^{4}}|v_{2}|^{2}$
$\displaystyle=\frac{1}{4\operatorname{Re}(u)^{2}}|v_{1}|^{2}+\frac{1}{4|u|^{2}}|v_{2}|^{2}.$

If $v\in L_{I_{u}}$ then its length is, not surprisingly, the hyperbolic
length in the hyperbolic half plane
$\mathbb{H}^{+}_{I_{u}}=\\{x+yI_{u}\,|\,x>0,y\in\mathbb{R}\\}$. Notice that
$C$ maps $\mathbb{B}_{I}$ to $\mathbb{H}^{+}_{I}$ for any $I\in\mathbb{S}$ and
it maps the totally geodesic submanifold $\mathbb{B}(\pi/2)$ to
$\mathbb{H}^{+}(\pi/2):=C\left(\mathbb{B}(\pi/2)\right)=\\{q\in\mathbb{H}^{+}\,|\,|q|=1\\}$
i.e. the right half of the three-dimensional unit sphere $\mathbb{S}^{3}$.
Then it is not difficult to verify that the isometry group of
$(\mathbb{H}^{+},h)$ is generated by the images under $C$ of isometry of
$(\mathbb{B},g)$ of type $(a)$, $(b)$ and $(c)$,

  * (a’)

linear maps preserving the positive real half-axis,

$q\mapsto q\lambda,$

with $\lambda>0$;

  * (b’)

isometries of the sphere of the imaginary units, which in polar coordinates
$r\geq 0$, $t\in[0,\pi/2)$, $I\in\mathbb{S}$ read as

$q=re^{tI}\mapsto T_{A}(q)=re^{tA(I)},$

where $A:\SS\to\SS$ is an isometry of $\SS$;

  * (c’)

the inversion in the three-dimensional unit (half) sphere

$q\mapsto\frac{1}{\overline{q}}.$

Acting on $\mathbb{H}^{+}(\pi/2)$ by means of isometries of type
$(a^{\prime})$ we obtain totally geodesic regions of the form
$\\{q\in\mathbb{H}^{+}\,|\,|q|=R\\}$ for $R>0$, that can be sliced in totally
geodesic two-dimensional submanifolds, corresponding to submanifolds of type
$D(t,C(J))$ in the ball case.

In this setting it is possible to introduce horocycles, i.e. hyperplanes of
points with constant real part,
$H_{c}=\\{q\in\mathbb{H}^{+}\,|\,\operatorname{Re}(q)=c\\}$ for some constant
$c>0$. They deserve the name of horocycles because their intersection with
each slice $L_{I}$ is a proper horocycle in the hyperbolic half plane
$\mathbb{H}^{+}_{I}$. Isometries of type $(a^{\prime})$ map horocycles one
into another.

If we restrict the metric $h$ to horocycles we obtain that the length of a
vector $v=v_{1}+v_{2}\in T_{u}H_{c}\cong\mathbb{R}I_{u}+L_{I_{u}}^{\perp}$
tangent to the horocycle $H_{c}$ at the point $u\in H_{c}$, can be written as

$|v|^{2}_{H_{c}}=\frac{1}{4c^{2}}|v_{1}|^{2}+\frac{1}{4(c^{2}+|\operatorname{Im}(u)|^{2})}|v_{2}|^{2}$

and the corresponding volume form at $u=c+x_{1}i+x_{2}j+x_{3}k$ is

$dVol_{H_{c}}(u)=\frac{dVol_{Euc}(u)}{8c(c^{2}+|\operatorname{Im}(u)|^{2})},$

(since the component in $L_{I_{u}}$ is one-dimensional) where
$dVol_{Euc}(u)=dx_{1}dx_{2}dx_{3}$ is the standard Euclidean volume element.
If we want to define a (non-degenerate) volume form
$dVol_{\partial\mathbb{H}^{+}}$ on the boundary $\partial\mathbb{H}^{+}$ of
the quaternionic right half space we can not directly take the limit of
$dVol_{H_{c}}$ as $c$ approaches $0$, we need indeed first to normalize it.
For any $u\in\partial\mathbb{H}^{+}$, we define

$dVol_{\partial\mathbb{H}^{+}}(u)\\!:=\\!\\!\lim_{c\to
0^{+}}c\left(dVol_{H_{c}}(u+c)\right)=\\!\lim_{c\to
0^{+}}\frac{dVol_{Euc}(u+c)}{8(c^{2}+|\operatorname{Im}(u+c)|^{2})}=\frac{dVol_{Euc}(u)}{8|\operatorname{Im}(u)|^{2}}=\frac{dVol_{Euc}(u)}{8|u|^{2}}.$
(14)

###  5.1 Hardy space on $\mathbb{H}^{+}$

We will show that, as in the case of the metric $g$ on $\mathbb{B}$, the
invariant metric $h$ on $\mathbb{H}^{+}$ introduced in the previous section
and in particular the corresponding volume form (14), is related with the
quaternionic Hardy space on the right half space $\mathbb{H}^{+}$. It is
possible to define the Hardy space $H^{2}(\mathbb{H}^{+})$ on $\mathbb{H}^{+}$
as the space of regular functions $f:\mathbb{H}^{+}\to\mathbb{H}$ of the form,

$f(q)=\int_{0}^{+\infty}e^{-\zeta q}F(\zeta)d\zeta,$

with $F:\mathbb{R}^{+}\to\mathbb{H}$, such that

$||f||^{2}_{H^{2}(\mathbb{H}^{+})}:=\int_{0}^{+\infty}|F(\zeta)|^{2}d\zeta<+\infty.$

With this definition, the reproducing kernel of $H^{2}(\mathbb{H}^{+})$ is a
function

$k(q,w)=k_{w}(q)=\int_{0}^{+\infty}e^{-\zeta q}G(\zeta)d\zeta$

where $G:\mathbb{R}^{+}\to\mathbb{H}$ is such that

$f(w)=\langle
f,k_{w}\rangle_{H^{2}(\mathbb{H}^{+})}=\int_{0}^{+\infty}\overline{G(\zeta)}F(\zeta)d\zeta=\int_{0}^{+\infty}e^{-\zeta
w}F(\zeta)d\zeta.$

Hence $G$ has to satisfy $\overline{G(\zeta)}=e^{-\zeta w}$ which implies
${G(\zeta)}=e^{-\zeta\bar{w}}$, i.e. that the kernel function is

$k_{w}(q)=\int_{0}^{+\infty}e^{-\zeta q}e^{-\zeta\bar{w}}d\zeta.$

To obtain a closed expression of $k_{w}(q)$, let first $q$ be a (positive)
real number. In this case $q$ commutes with all points in $\mathbb{H}^{+}$ and
we can write

$k_{w}(q)=\int_{0}^{+\infty}e^{-\zeta(q+\bar{w})}d\zeta=\frac{1}{q+\overline{w}}.$

Consider now the function $q\mapsto(q+\overline{w})^{-*}$ (here the regular
reciprocal is defined with a slight generalization of Definition 2.10, see
[12]). This function is regular and it coincides with
$q\mapsto(q+\overline{w})^{-1}$ on real numbers. Thanks to the Identity
Principle for regular functions, Theorem 1.12 in [12], we obtain that the
reproducing kernel is
$k_{w}(q)=(w+\overline{q})^{-*}=\int_{0}^{+\infty}e^{-\zeta
w}e^{-\zeta\bar{q}}d\zeta$.

Another way to obtain the reproducing kernel on $H^{2}(\mathbb{H}^{+})$ is the
following.

######  Proposition 5.1.

Denote by $k_{H^{2}(\mathbb{B})}$ and by $k_{H^{2}(\mathbb{H}^{+})}$ the
reproducing kernels of the Hardy space on the unit ball $H^{2}(\mathbb{B})$
and on the right half-space $H^{2}(\mathbb{H}^{+})$ respectively. Let
$C:\mathbb{B}\to\mathbb{H}^{+}$ be the Cayley map, $C(q)=(1-q)^{-1}(1+q)$. For
any $z,w\in\mathbb{H}^{+}$, the function
$k_{H^{2}(\mathbb{B})}(C^{-1}(w),C^{-1}(z))$ is a rescaling of the reproducing
kernel of $H^{2}(\mathbb{H}^{+})$:

$k_{H^{2}(\mathbb{B})}(C^{-1}(w),C^{-1}(z))=\frac{1}{2}(1+z)k_{H^{2}(\mathbb{H}^{+})}(w,z)(1+\overline{w}).$

###### Proof.

The map $C^{-1}$, having real coefficients, is slice preserving. Hence, we can
compose $k_{H^{2}(\mathbb{B})}$ with $C^{-1}$ preserving (left) regularity in
the first variable and (right) “anti-regularity” in the second one. We have,
then,

$\displaystyle
k_{H^{2}(\mathbb{B})}(C^{-1}(z),C^{-1}(w))=(1-q\overline{C^{-1}(w)})_{|_{q=C^{-1}(z)}}^{-*}$
$\displaystyle=\left(1-2C^{-1}(z)\operatorname{Re}(C^{-1}(w))+C^{-1}(z)^{2}|C^{-1}(w)|^{2}\right)^{-1}\left(1-C^{-1}(z)C^{-1}(w)\right)$
$\displaystyle=(1+z)^{2}|1+w|^{2}\left(|1+w|^{2}(1+z)^{2}-2(1-z^{2})(1-|w|^{2})+(1-z)^{2}|1-w|^{2}\right)^{-1}$
$\displaystyle\hskip
113.81102pt\cdot(1+z)^{-1}\left((1+z)(1+w)-(1-z)(1-w)\right)(1+w)^{-1}$
$\displaystyle=(1+z)\frac{1}{4}\left(z^{2}+2z\operatorname{Re}(w)+|w|^{2}\right)^{-1}2\left(z+w\right)(1+\overline{w})$
$\displaystyle=\frac{1}{2}(1+z)\left(z+\overline{w}\right)^{-*}(1+\overline{w})=\frac{1}{2}(1+z)k_{H^{2}(\mathbb{H}^{+})}(w,z)(1+\overline{w}).$

∎

Now we want to show that the volume form on $\partial\mathbb{H}^{+}$ obtained
in (14) is the natural volume form for the Hardy space
$H^{2}(\mathbb{H}^{+})$. In fact, let $f(q)=\int_{0}^{+\infty}e^{-\zeta
q}F(\zeta)d\zeta\in H^{2}(\mathbb{H}^{+})$. For any $I\in\mathbb{S}$, we can
decompose the function $F$ as $F(\zeta)=F_{1}(\zeta)+F_{2}(\zeta)J$ where $J$
is an imaginary unit orthogonal to $I$ and $F_{1},F_{2}:\mathbb{R}\to L_{I}$.
It is possible to prove (see [2]) that functions in $H^{2}(\mathbb{H}^{+})$
have limit at the boundary for almost any point
$yI\in\partial\mathbb{H}^{+}=\\{vJ\ |\ v>0,J\in\mathbb{S}\\}$. If we denote by
$dA_{\mathbb{S}}$ the usual surface element of the unit two-dimensional sphere
$\mathbb{S}$, thanks to equation (14) and to the orthogonality of $I$ and $J$,
we can write

$\displaystyle\int_{\partial\mathbb{H}^{+}}|f(yI)|^{2}dVol_{\partial\mathbb{H}^{+}}(yI)=\int_{\partial\mathbb{H}^{+}}|f(yI)|^{2}\frac{dVol_{Euc}(yI)}{8y^{2}}$
$\displaystyle=\int_{0}^{+\infty}\left(\int_{\mathbb{S}}\left|\int_{0}^{+\infty}e^{-\zeta
yI}F(\zeta)d\zeta\right|^{2}\frac{y^{2}dA_{\mathbb{S}}(I)}{8y^{2}}\right)dy$
$\displaystyle=\frac{1}{8}\int_{\mathbb{S}}\left(\int_{0}^{+\infty}\left|\int_{0}^{+\infty}e^{-\zeta
yI}\left(F_{1}(\zeta)+F_{2}(\zeta)J\right)d\zeta\right|^{2}dy\right)dA_{\mathbb{S}}(I)$
$\displaystyle=\frac{1}{8}\int_{\mathbb{S}}\left(\int_{0}^{+\infty}\left|\int_{0}^{+\infty}e^{-\zeta
yI}F_{1}(\zeta)d\zeta+\int_{0}^{+\infty}e^{-\zeta
yI}F_{2}(\zeta)Jd\zeta\right|^{2}dy\right)dA_{\mathbb{S}}(I)$
$\displaystyle=\frac{1}{8}\int_{\mathbb{S}}\left(\int_{0}^{+\infty}\left|\int_{0}^{+\infty}e^{-\zeta
yI}F_{1}(\zeta)d\zeta\right|^{2}dy+\int_{0}^{+\infty}\left|\int_{0}^{+\infty}e^{-\zeta
yI}F_{2}(\zeta)d\zeta\right|^{2}dy\right)dA_{\mathbb{S}}(I)$
$\displaystyle=\frac{2\pi}{8}\int_{\mathbb{S}}\left(\int_{0}^{+\infty}\left|F_{1}(\zeta)\right|^{2}d\zeta+\int_{0}^{+\infty}\left|F_{2}(\zeta)\right)|^{2}d\zeta\right)dA_{\mathbb{S}}(I)$

where the last equality is due to the classical Plancherel Theorem. Therefore,
thanks again to the orthogonality of $I$ and $J$,

$\int_{\partial\mathbb{H}^{+}}|f(yI)|^{2}dVol_{\partial\mathbb{H}^{+}}(yI)=\frac{\pi}{4}\int_{\mathbb{S}}\left(\int_{0}^{+\infty}\left|F(\zeta)\right|^{2}d\zeta\right)dA_{\mathbb{S}}(I)=\pi^{2}||f||^{2}_{H^{2}(\mathbb{H}^{+})}.$

###  5.2 A bilateral estimate for the distance and an application to inner
functions

In the right half space model it is easier to prove a bilateral estimate for
the distance associated with the invariant metric $h$. Fix a imaginary unit
$I_{0}$ and define the projection

$\pi:x+yI\mapsto x+yI_{0},$ (15)

with $x$ real and $y\geq 0$. Let $d_{hyp}$ be hyperbolic distance in
$\mathbb{H}_{I_{0}}^{+}=\\{x+yI_{0}:\ x>0,\ y\in\mathbb{R}\\}$: $d_{hyp}$ is
the distance associated with the Riemannian metric tensor
$ds_{hyp}^{2}=(dx^{2}+dy^{2})/(4x^{2})$. Let now $d_{\SS}$ be the usual
spherical distance on the unit two-dimensional sphere $\SS$, associated with
the metric tensor $ds^{2}_{\mathbb{S}}$. Then, the metric tensor associated
with $h$ can be decomposed as

$ds_{h}^{2}=ds^{2}_{hyp}+\frac{y^{2}}{4(x^{2}+y^{2})}ds^{2}_{\mathbb{S}}.$

######  Theorem 5.2.

Let $q_{j}=x_{j}+y_{j}I_{j}$, $j=1,2$, be points in $\mathbb{H}^{+}$:
$x_{j}>0$, $y_{j}\geq 0$. The following estimate for the distance function
$d_{h}$ associated with the metric $h$ holds:

$d_{h}(q_{1},q_{2})\approx
d_{hyp}(\pi(q_{1}),\pi(q_{2}))+\min\left\\{\frac{y_{j}}{|q_{j}|}:\
j=1,2\right\\}d_{\SS}(I_{1},I_{2}),$

where $\approx$ means that we have a lower and an upper estimate for the right
hand side in terms of the left hand side, with multiplicative constants
$C_{1},C_{2}$ independent of $q_{1},q_{2}$.

###### Proof.

We may suppose that $y_{1}/|q_{1}|\leq y_{2}/|q_{2}|$.

The upper estimate is elementary. Let $\gamma$ be a curve going from $q_{1}$
to $x_{1}+y_{1}I_{2}\in\mathbb{H}_{I_{2}}^{+}$ leaving $x=x_{1}$ and $y=y_{1}$
fixed, and varying the imaginary unit $I$ only. Suppose, more, that $I$ varies
along a geodesic on $\SS$, which joins $I_{1}$ and $I_{2}$. Then,

$\operatorname{length}(\gamma)=\int_{\gamma}\frac{y_{1}}{2|q_{1}|}ds_{\mathbb{S}}=\frac{y_{1}}{2|q_{1}|}d_{\SS}(I_{1},I_{2}).$

Let now $\delta$ be a hyperbolic geodesic in $L_{I_{2}}$, joining
$x_{1}+y_{1}I_{2}$ and $q_{2}$:
$\operatorname{length}(\delta)=d_{hyp}(\pi(q_{1}),\pi(q_{2}))$, which proves
the estimate.

The lower estimate is more delicate. Let $\gamma$ be a curve in
$\mathbb{H}^{+}$ joining $q_{1}$ and $q_{2}$. Then,

$\operatorname{length}(\gamma)=\int_{\gamma}ds_{h}\geq\int_{\pi(\gamma)}ds_{hyp}\geq
d_{hyp}(\pi(q_{1}),\pi(q_{2})).$ (16)

We have then to show that

$\int_{\gamma}ds_{h}\gtrsim\frac{y_{1}}{|q_{1}|}d_{\SS}(I_{1},I_{2}).$ (17)

Since the right hand side of (17) is bounded, and we have already proved (16),
it suffices to show that (17) holds when $d_{hyp}(\pi(q_{1}),\pi(q_{2}))\leq
1$. By elementary hyperbolic geometry, see the “sixth model” in [7], and using
the fact that dilations $p\mapsto\lambda p$ are isometric for $\lambda>0$, we
can assume that $\pi(q_{1})$ and $\pi(q_{2})$ both lie in the square
$Q_{n}=\\{x+yI_{0}:\ 1\leq x\leq 2,\ n\leq y\leq n+1\\}\subset L_{I_{0}}$, for
some integer $n\geq 0$. Consider now $q_{3}=x_{3}+y_{3}I_{3}$, $y_{3}\geq 0$
the point along $\gamma$ which minimizes $y_{3}/|q_{3}|$. We can assume that
$\pi(\gamma)$ (hence, $\pi(q_{3})$) is contained in
$\tilde{Q}_{n}=\\{z=x+yI_{0}:\ x>0,\ y\geq 0,\ 1/2\leq x\leq 2,\ n-1/2\leq
y\leq n+3/2\\}$, otherwise $\operatorname{length}(\gamma)\geq 1$ (which would
imply the estimate (17) we are proving). Let $t\geq 0$ be the angle between
the positive real half axis $\mathbb{R}^{+}$ and the half line originating at
$0$ and passing through $\pi(q_{3})$. For $j=1,2,3$:

$t_{j}\approx\sin(t_{j})=y_{j}/|q_{j}|.$

We have two cases. Either $y_{3}/|q_{3}|\geq 1/2\cdot y_{1}/|q_{1}|$, but then
we are done because

$\int_{\gamma}\frac{y}{2|q|}ds^{2}_{\mathbb{S}}\geq\frac{y_{1}}{2|q_{1}|}d_{\SS}(I_{1},I_{2}).$

Or $y_{3}/|q_{3}|\leq 1/2\cdot y_{1}/|q_{1}|$. Then $n=0$, and

$\displaystyle\int_{\gamma}ds_{h}$ $\displaystyle\geq$
$\displaystyle\operatorname{length}(\pi(\gamma))\gtrsim\max(|\pi(z_{1})-\pi(z_{3})|,|\pi(z_{2})-\pi(z_{3})|)\geq|\pi(z_{1})-\pi(z_{3})|$
$\displaystyle\gtrsim$ $\displaystyle
y_{1}\gtrsim\frac{y_{1}}{|z_{1}|}d_{\SS}(I_{1},I_{2}).$

Overall,
$\int_{\gamma}ds_{h}\gtrsim\frac{y_{1}}{|z_{1}|}d_{\SS}(I_{1},I_{2})$, as
wished. ∎

Changing coordinates from the right half plane to the ball, we have the same
bilateral estimate in the ball model.

######  Corollary 5.3.

Let $d_{g}$ be the invariant distance associated with the metric $g$ in the
ball model and let $q_{1},q_{2}$ be points of $\mathbb{B}$. If $\pi$ is
defined as in (15), then:

$d_{g}(q_{1},q_{2})\approx
d_{hyp}(\pi(q_{1}),\pi(q_{2}))+\min\left\\{\frac{y_{j}}{|1-q_{j}^{2}|}:\
j=1,2\right\\}d_{\SS}(I_{1},I_{2}).$

A regular function $f:\mathbb{B}\to\mathbb{H}$ is inner if (i) it maps
$\mathbb{B}$ into $\mathbb{B}$; (ii) the limit as $r\to 1$ of $f$ along the
radius $r\mapsto ru$ exists for $a.e.$ $u$ in $\partial\mathbb{B}$ and it has
unitary norm.

######  Theorem 5.4.

Let $f:\mathbb{B}\to\mathbb{B}$ be an inner function. Then, $f$ is Lipschitz
with respect to the metric $g$ if and only if it is slice preserving. In this
case, it is a contraction.

It is well known (see [10]) that regular, bounded functions have radial limits
along almost all radii,

$f(e^{tI}):=\lim_{r\to 1}f(re^{tI})$

exists for $a.e.$ $(t,I)\in[0,\pi]\times\SS$. We start with a Lemma which
might have independent interest; for instance, it provides a different route
to prove the classification of the isometries for the metric $g$.

######  Lemma 5.5.

If $\varphi:\mathbb{B}\to\mathbb{B}$ is Lipschitz with respect to the metric
$g$ and

$\lim_{r\to 1}\varphi(re^{tI_{1}})=e^{sJ_{1}}\in\partial\mathbb{B}$ (18)

exists, with $s\in[0,\pi]$ and $J_{1}\in\SS$; then for each $I_{2}$ in $\SS$,
if the limit $\lim_{r\to 1}\varphi(re^{I_{2}t})$ exists, then

$\lim_{r\to 1}\varphi(re^{tI_{2}})=e^{sJ_{2}}$ (19)

for some $J_{2}$ in $\SS$. The values of $t$ and $s$ in (19) are the same as
in (18).

###### Proof.

Let $u_{j}=e^{tI_{j}}$, with the same $t\in[0,\pi]$. By Lipschitz continuity,

$\displaystyle d(\varphi(ru_{1}),\varphi(ru_{2}))$ $\displaystyle\lesssim$
$\displaystyle d(ru_{1},ru_{2})$ (20) $\displaystyle\approx$
$\displaystyle\frac{rt|I_{1}-I_{2}|}{(1-r)+rt}\lesssim|I_{1}-I_{2}|$ (21)
$\displaystyle\leq$ $\displaystyle 1.$ (22)

By the lower estimate in Corollary 5.3 and (20),

$d_{hyp}\left(\pi\left(\varphi(ru_{1})\right),\pi\left(\varphi(ru_{2})\right)\right)\lesssim
1.$

But this and elementary hyperbolic geometry imply that, if $\lim_{r\to
1}\varphi(ru_{1})=e^{sJ_{1}}$, then the limit $\lim_{r\to
1}\pi\left(\varphi(ru_{2})\right)=L$ exists and $L=e^{sI_{0}}$ (recall that
$\pi:\mathbb{B}\to L_{I_{0}}$). Since $\lim_{r\to 1}\varphi(re^{tI_{2}})=L$
exists by hypothesis and $\pi$ is continuous, it must be $\pi(L)=e^{sI_{0}}$,
then $L=e^{sJ_{2}}$ for some $J_{2}$ in $\SS$.

∎

The statement of Lemma 5.5 can be sharpened in several ways. For instance, the
Lipschitz assumption might be weakened to a sub-exponential growth assumption.

We proceed with the proof of Theorem 5.4.

###### Proof of Theorem 5.4.

Being inner, $f$ has boundary limits along radii $r\mapsto re^{tI}$ for $a.e.$
$I$ in $\SS$ and $t$ in $[0,\pi]$. We write for such couples of $(t,I)$:
$f(e^{tI}):=\lim_{r\to 1}f(re^{tI})$. We can assume without loss of generality
that the limit exists for two antipodal imaginary units $L$ and $-L$, and
hence, in view of the Representation Formula 2.5, for any $L\in\mathbb{S}$. If
$f$ is regular and Lipschitz with respect to the distance $d_{g}$, thanks on
the one hand to the Representation Formula 2.5, on the other hand to Lemma
5.5, we have that for any $I\in\mathbb{S}$

$b(t)+Ic(t)=f(e^{tI})=e^{s(t)J(s,I)}$

where $b(t),c(t)\in\mathbb{H}$, and $s(t)\in[0,2\pi]$ and
$J(t,I)\in\mathbb{S}$. Then
$\operatorname{Re}(f(e^{tI}))=\operatorname{Re}(f(e^{tL}))$ for any
$L\in\mathbb{S}$ and in particular for $L=-I$, which gives

$\operatorname{Re}(b(t))-\langle
I,c(t)\rangle=\operatorname{Re}(b(t)+Ic(t))=\operatorname{Re}(b(t)-Ic(t))=\operatorname{Re}(b(t))+\langle
I,c(t)\rangle$

(where $\langle\cdot,\cdot\rangle$ denotes the standard scalar product in
$\mathbb{R}^{4}$). Since $I$ is any imaginary unit, we necessarily have that
$c(t)\in\mathbb{R}$.

Also, comparing imaginary parts, for any $L_{1},L_{2}\in\mathbb{S}$ we have
$|\operatorname{Im}(f(e^{tL_{1}}))|=|\operatorname{Im}(f(e^{tL_{2}}))|$. Then,
if $b=b_{0}+b_{1}K$ with $b_{0},b_{1}\in\mathbb{R}$, $K\in\mathbb{S}$
(omitting the dependence on $t$), when $L_{1}=K$ and $L_{2}=-K$ we get

$|b_{1}+c|=|\operatorname{Im}(b_{0}+b_{1}K+cK)|=|\operatorname{Im}(f(e^{tK}))|=|\operatorname{Im}(f(e^{-tK}))|=|\operatorname{Im}(b_{0}+b_{1}K-cK)|=|b_{1}-c|.$

Therefore almost every $t\in[0,\pi]$ belongs to $D\cup E$:

$D=\\{t:\ c(t)=0\\},\ E=\\{t:\ b_{1}(t)=0\\}.$

Consider first the case when $t\in D$ holds $a.e.$. Then $f(e^{tI})=b(t)$ for
almost every $t$. Since boundary values uniquely identify $f$ (see [10]) and
by invariance under rotations of $\SS$, we deduce that

$f(re^{tI})=\Phi(r,t),$

for some function $\Phi$. In particular, $f$ can not be open
($dim(f(\mathbb{B}))\leq 2$), hence (see Theorem 7.4 in [12]) it must be
constant; thus it is not inner. Then $E$ has positive measure. For $t$ in $E$,

$b(t)+Ic(t)=f(e^{tI})=e^{s(t)J(t,I)}$

with $b$ and $c$ real valued, hence $J=I$:

$f(e^{tI})=e^{s(t)I}$ (23)

for $t$ in $F$. By the Splitting Lemma 2.7, if $J\perp I$ is fixed in $\SS$,
then there are holomorphic functions $F,G$ on $\mathbb{B}_{I}$ such that

$f(re^{\tau I})=F(re^{\tau I})+G(re^{\tau I})J.$

By (23), $G(e^{tI})=0$ for $t$ in $E$. Since $E$ has positive measure, this
implies that $G$ vanishes identically and hence $f(re^{\tau I})=F(re^{\tau
I})$ for all $0\leq r<1$ and $0\leq\tau\leq\pi$. That is, $f$ is slice
preserving.

We have to verify that $f$ is a contraction with respect to the metric $g$,
and this can be verified at the infinitesimal level. Let $q$ be a point in a
fixed slice $\mathbb{B}\cap L_{I}$. (i) Since $f$ is slice preserving, its
restriction to $\mathbb{B}\cap L_{I}$ is an inner function in the one
dimensional sense, hence it is a contraction of the Poincaré-hyperbolic metric
on $\mathbb{B}\cap L_{I}$. (ii) On the other hand, preserving the slices, $f$
acts isometrically in the $\SS$ variables, with respect to the spherical
metric on $\SS$. (iii) Now, the space tangent to $\mathbb{B}\cap L_{I}$ at $q$
and the space tangent to $\operatorname{Re}q+\SS$ at $q$ form an orthogonal
decomposition, with respect to the metric $g$, of the space tangent to
$\mathbb{B}$. From the expression for $g$ given in (8) and facts (i)-(iii) one
easily deduces that $g$ is a contraction. ∎

## References

  * [1] D. Alpay, F. Colombo, I. Sabadini, Schur functions and their realizations in the slice hyperholomorphic setting, Integral Equations Operator Theory 72 (2012), 25–289.
  * [2] D. Alpay, F. Colombo, I. Lewkowicz, I. Sabadini, Realizations of slice hyperholomorphic generalized contractive and positive functions, Preprint arXiv:1310.1035v1 [math.CV] (2013).
  * [3] N. Arcozzi, R. Rochberg, E. Sawyer, B. D. Wick, Distance functions for reproducing kernel Hilbert spaces, Function spaces in modern analysis, 25–53, Contemp. Math., 547, Amer. Math. Soc., Providence, RI, 2011.
  * [4] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68, (1950) 337–404.
  * [5] C. Bisi, G. Gentili, Möbius transformations and the Poincaré distance in the quaternionic setting, Indiana Univ. Math. J., 58 (2009), 2729–2764.
  * [6] C. Bisi, C. Stoppato, Regular vs. classical Möbius transformations of the quaternionic unit ball, Advances in hypercomplex analysis, Springer INdAM Ser. 1, Springer, Milan, 2013, 1–13.
  * [7] J. W. Cannon, W. J. Floyd, R. Kenyon, W. R. Parry, Hyperbolic geometry, in Flavors of geometry, Math. Sci. Res. Inst. Publ., 31, Cambridge Univ. Press, Cambridge (1997), 59–115,.
  * [8] F. Colombo, J. O. Gonzalez-Cervantes, M. E. Luna-Elizarraras, I. Sabadini, M. V. Shapiro, On two approaches to the Bergman theory for slice regular functions, Advances in hypercomplex analysis, Springer INdAM Ser. 1, Springer, Milan, 2013, 39–54.
  * [9] M. J. Cowen, R. G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), 187–261.
  * [10] C. de Fabritiis, G. Gentili, G. Sarfatti, Quaternionic Hardy spaces, Preprint, www.math.unifi.it/users/sarfatti/Hardy.pdf, (2013).
  * [11] M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.
  * [12] G. Gentili, C. Stoppato, D. C. Struppa, Regular functions of a quaternionic variable, Springer Monographs in Mathematics, Springer, Berlin-Heidelberg, 2013.
  * [13] G. Gentili, D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math., 216 (2007), 279–301.
  * [14] R. Ghiloni, V. Moretti, A. Perotti, Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25 (2013).
  * [15] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Based on the 1981 French original, translated from the French by Sean Michael Bates. Progress in Mathematics, 152, Birkh user Boston, Inc., Boston, MA, 1999.
  * [16] J. E. McCarthy, Boundary values and Cowen-Douglas curvature, J. Funct. Anal. 137 (1996), 1–18.
  * [17] B. O’Neill, Semi-Riemannian geometry, with applications to relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983.
  * [18] C. Stoppato, Regular Moebius transformations of the space of quaternions, Ann. Global Anal. Geom. 39 (2011), 387–401.

Nicola Arcozzi
Dipartimento di Matematica, Università di Bologna
Piazza di Porta San Donato 5, 40126 Bologna, Italy<EMAIL_ADDRESS>
Giulia Sarfatti
Dipartimento di Matematica, Università di Bologna
Piazza di Porta San Donato 5, 40126 Bologna, Italy<EMAIL_ADDRESS>