# A characterization of the bidisc by a subgroup of its automorphism group

Anindya Biswas  Department of Mathematics, Indian Institute of Science,
Bangalore 560012, India<EMAIL_ADDRESS>and  Anwoy Maitra  Harish-
Chandra Research Institute, Chhatnag Road, Jhunsi, Prayagraj 211019, India.
<EMAIL_ADDRESS>(Corresponding author)

###### Abstract.

We make a connection between the structure of the bidisc and a distinguished
subgroup of its automorphism group. The automorphism group of the bidisc, as
we know, is of dimension six and acts transitively. We observe that it
contains a subgroup that is isomorphic to the automorphism group of the open
unit disc and this subgroup partitions the bidisc into a complex curve and a
family of strongly pseudo-convex hypersurfaces that are non-spherical as CR-
manifolds. Our work reverses this process and shows that any $2$-dimensional
Kobayashi-hyperbolic manifold whose automorphism group (which is known, from
the general theory, to be a Lie group) has a $3$-dimensional subgroup that is
non-solvable (as a Lie group) and that acts on the manifold to produce a
collection of orbits possessing essentially the characteristics of the
concretely known collection of orbits mentioned above, is biholomorphic to the
bidisc.

The distinguished subgroup is interesting in its own right. It turns out that
if we consider any subdomain of the bidisc that is a union of a proper sub-
collection of the collection of orbits mentioned above, then the automorphism
group of this subdomain can be expressed very simply in terms of this
distinguished subgroup.

###### Key words and phrases:

Bidisc, automorphism group, classification of hyperbolic manifolds, subdomains
of the bidisc

###### 2010 Mathematics Subject Classification:

Primary 32M05, Secondary 32V15, 32V40

##  1\. Introduction

This paper is concerned with presenting a characterization of the bidisc.
Namely, we will pick out the bidisc from among all 2-dimensional Kobayashi-
hyperbolic complex manifolds by making some assumptions about the automorphism
group of such a manifold and about its action on the manifold.

There has been work—see [13]—characterizing the bidisc by picking it out from
among all bounded domains in $\mathbb{C}^{2}$ by making some extrinsic
assumptions about the action of the automorphism group of the domain on the
domain (the assumptions are extrinsic in that they deal with the convergence
of certain sequences to boundary points). In our work, we will only make
intrinsic assumptions about the automorphism group and its action on the
manifold. Our work is heavily dependent on the seminal work [8], in which all
2-dimensional Kobayashi-hyperbolic complex manifolds with (real) 3-dimensional
automorphism group have been classified. The current work is also an outgrowth
of the earlier work [2] of the two present authors (joint with T.
Bhattacharyya).

The bidisc (along with the Euclidean unit ball in $\mathbb{C}^{2}$) is a pre-
eminent model domain that has been studied intensively since the earliest days
of several complex variables. It is, therefore, of interest to have
characterizations of it. Roughly speaking, our main result says that any
2-dimensional Kobayashi-hyperbolic complex manifold whose automorphism group
(which is known to be a Lie group) is assumed to have a certain non-solvable
Lie _subgroup_ under the action of which all the orbits, except only one, are
assumed to be strongly pseudoconvex hypersurfaces, and the sole remaining
orbit is assumed to be a complex curve, must be biholomorphic to the bidisc if
one makes the further assumption that these hypersurfaces are of a specific
type. To be more precise, we make the assumption that these hypersurfaces are
CR-equivalent to certain standard hypersurfaces that occur in E. Cartan’s
famous classification of connected homogeneous strongly pseudoconvex
3-dimensional CR manifolds [3] (see also [6]). Equivalently, we make the
assumption that the hypersurfaces are CR-equivalent to certain hypersurfaces
(see (2.2) and Proposition 2.1) one obtains when a particular distinguished
subgroup (see (2.1)) of $Aut(\mathbb{D}^{2})$ acts on $\mathbb{D}^{2}$. Along
the way, we also obtain characterizations of some distinguished sub-domains of
$\mathbb{D}^{2}$ (see Theorem 2.3).

A noteworthy feature of our work is that we only make assumptions about some
subgroup of the automorphism group of the initially arbitrary manifold that we
start out with. To the best of our knowledge, this is a new approach.

The plan of this work is as follows: in Section 2, we introduce the
distinguished subgroup of $Aut(\mathbb{D}^{2})$ and the distinguished sub-
domains of $\mathbb{D}^{2}$ mentioned above, and connect them to certain model
domains exhibited in [8] and their automorphism groups. After that we
investigate the action of some subgroups of $Aut(\mathbb{B}_{2})$ on
$\mathbb{B}_{2}$, where $\mathbb{B}_{2}$ denotes the Euclidean unit ball in
$\mathbb{C}^{2}$ (this is needed for the proof of our main theorem). Finally,
in Section 3, we prove our main result characterizing $\mathbb{D}^{2}$.

##  2\. Preliminary results

###  2.1. Some domains in $\mathbb{D}^{2}$ and their automorphism groups

We start with $\mathbb{D}^{2}$ and its automorphism group which is given by

$\displaystyle
Aut(\mathbb{D}^{2})=\\{(\varphi_{1}\times\varphi_{2}),(\varphi_{1}\times\varphi_{2})\circ\sigma:\varphi_{1},\varphi_{2}\in
Aut(\mathbb{D})\\},$

where
$(\varphi_{1}\times\varphi_{2})(z_{1},z_{2})=(\varphi_{1}(z_{1}),\varphi_{2}(z_{2}))$
and $\sigma(z_{1},z_{2})=(z_{2},z_{1})$. This can be found in [12]. In [2], a
closed subgroup of $Aut(\mathbb{D}^{2})$ has been identified and its action on
$\mathbb{D}^{2}$ has been observed. The subgroup is

(2.1) $\displaystyle G_{\mathbb{D}}=\\{\Phi_{\varphi}:\varphi\in
Aut(\mathbb{D})\\},$

where $\Phi_{\varphi}(z_{1},z_{2})=(\varphi(z_{1}),\varphi(z_{2}))$. This
group does not act transitively and its orbits are

(2.2)
$\displaystyle\mathscr{F}_{a}=\Big{\\{}(z_{1},z_{2})\in\mathbb{D}^{2}:\Big{|}\frac{z_{1}-z_{2}}{1-\overline{z_{1}}z_{2}}\Big{|}=a\Big{\\}},\,\,a\in[0,1).$

From [2] we know that $\mathscr{F}_{0}$ is a complex curve and
$\mathscr{F}_{a}$ is a strongly pseudoconvex hypersurface for each
$a\in(0,1)$.

In [5] and [8], the author listed some collections of domains which are given
by

(2.3) $\displaystyle\mathcal{D}^{(2)}_{s,t}=$
$\displaystyle\\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}:s<|z_{1}|^{2}+|z_{2}|^{2}-|z_{3}|^{2}<t,z_{1}^{2}+z_{2}^{2}-z_{3}^{2}=1,$
$\displaystyle Im(z_{2}(\overline{z_{1}}+\overline{z_{3}}))>0\\},1\leq
s<t\leq\infty$

and

(2.4) $\displaystyle\mathcal{D}^{(2)}_{s}=$
$\displaystyle\\{(1:z_{1}:z_{2}:z_{3})\in\mathbb{CP}^{3}:s<|z_{1}|^{2}+|z_{2}|^{2}-|z_{3}|^{2},z_{1}^{2}+z_{2}^{2}-z_{3}^{2}=1,$
$\displaystyle
Im(z_{2}(\overline{z_{1}}+\overline{z_{3}}))>0\\}\cup\\{(0:z_{1}:z_{2}:z_{3})\in\mathbb{CP}^{3}:z_{1}^{2}+z_{2}^{2}-z_{3}^{2}=0,$
$\displaystyle Im(z_{2}(\overline{z_{1}}+\overline{z_{3}}))>0\\},1<s.$

Here
$\mathcal{O}^{(2)}=\\{(0:z_{1}:z_{2}:z_{3})\in\mathbb{CP}^{3}:z_{1}^{2}+z_{2}^{2}-z_{3}^{2}=0,Im(z_{2}(\overline{z_{1}}+\overline{z_{3}}))>0\\}$
is a complex curve and we have
$\mathcal{D}^{(2)}_{s}=\mathcal{D}^{(2)}_{s,\infty}\cup\mathcal{O}^{(2)}$.

The connected components containing the identity of the automorphism groups of
the domains (2.3) and (2.4) are all equal to $SO_{2,1}(\mathbb{R})^{0}$ (the
connected component of $SO_{2,1}(\mathbb{R})$ containing the
identity—hereinafter, for $K$ an arbitrary topological group, we shall use
$K^{0}$ to denote the connected component of $K$ containing the identity) with
the following action

(2.5) $\displaystyle\begin{pmatrix}z_{1}\\\ z_{2}\\\ z_{3}\end{pmatrix}\mapsto
A\begin{pmatrix}z_{1}\\\ z_{2}\\\ z_{3}\end{pmatrix}$

and

(2.6) $\displaystyle\begin{pmatrix}s\\\ z_{1}\\\ z_{2}\\\
z_{3}\end{pmatrix}\mapsto\begin{pmatrix}1&\mathbf{0}\\\
\mathbf{0}&A\end{pmatrix}\begin{pmatrix}s\\\ z_{1}\\\ z_{2}\\\
z_{3}\end{pmatrix},\;s=0,1,$

respectively. We write $\mathcal{R}^{(1)}$ to denote this group
($SO_{2,1}(\mathbb{R})^{0}$ with the action described above). The
$\mathcal{R}^{(1)}$-orbits of (2.3) are given by

(2.7)
$\displaystyle\eta^{(2)}_{\alpha}=\Bigg{\\{}(z_{1},z_{2},z_{3})\in\mathcal{D}^{(2)}_{s,t}:|z_{1}|^{2}+|z_{2}|^{2}-|z_{3}|^{2}=\sqrt{\frac{\alpha+1}{2}}\Bigg{\\}},s<\alpha<t.$

It is known from [6] that the hypersurfaces $\eta^{(2)}_{\alpha}$ are strongly
pseudoconvex, non-spherical and pairwise CR-nonequivalent. The
$\mathcal{R}^{(1)}$-orbits of (2.4) are $\eta^{(2)}_{\alpha}$, $\alpha>s$,
together with the complex curve $\mathcal{O}^{(2)}$.

Now we shall connect these domains $\mathcal{D}^{(2)}_{s,t}$ and
$\mathcal{D}^{(2)}_{s}$ with domains in $\mathbb{D}^{2}$. Consider the map
$J:\mathbb{D}^{2}\rightarrow\mathbb{CP}^{3}$ defined by

(2.8) $\displaystyle J(z,w)=(z-w:1-zw:i(1+zw):-i(z+w)).$

In the article [2], the authors showed that $J$ is a biholomorphism from
$\mathbb{D}^{2}$ onto $\mathcal{D}^{(2)}_{1}$. It maps
$D=\\{(z,z):z\in\mathbb{D}\\}$ onto $\mathcal{O}^{(2)}$. From this map we
derive $H:\mathbb{D}^{2}-D\rightarrow\mathbb{C}^{3}$ defined by

(2.9) $\displaystyle
H(z,w)=\Big{(}\frac{1-zw}{z-w},i\frac{1+zw}{z-w},-i\frac{z+w}{z-w}\Big{)}.$

######  Proposition 2.1.

$H$ maps $\mathbb{D}^{2}-D$ onto $\mathcal{D}^{(2)}_{1,\infty}$
biholomorphically. Moreover, for any $a\in(0,1)$, $H$ gives rise to a CR-
isomorphism between $\mathscr{F}_{a}$ and $\eta^{(2)}_{\alpha}$ where
$\alpha=\frac{8}{a^{4}}-\frac{8}{a^{2}}+1$.

###### Proof.

The proof that $H$ is a biholomorphism between $\mathbb{D}^{2}-D$ and
$\mathcal{D}^{(2)}_{1,\infty}$ follows from the observations that the $J$
above is a biholomorphism that maps $D=\\{(z,z):z\in\mathbb{D}\\}$ onto
$\mathcal{O}^{(2)}$ and that
$\mathcal{D}^{(2)}_{1}=\mathcal{D}^{(2)}_{1,\infty}\cup\mathcal{O}^{(2)}$. The
remaining part of the proposition follows from the observation that $H$ maps
$\mathscr{F}_{a}$ onto $\eta^{(2)}_{\alpha}$, where $\alpha$ and $a$ are
related as described above. ∎

The maps $J$ and $H$ open up a way to finding distinguished subdomains of
$\mathbb{D}^{2}$ and $\mathbb{D}^{2}-D$. For $r,s,t$ with $1\leq
s<t\leq\infty$ and $1<r$, consider the domains $J^{-1}(\mathcal{D}^{(2)}_{r})$
and $H^{-1}(\mathcal{D}^{(2)}_{s,t})$ in $\mathbb{D}^{2}$. From the
definitions of $\mathcal{D}^{(2)}_{r}$ and $\mathcal{D}^{(2)}_{s,t}$, it is
easy to see that

$\displaystyle
H^{-1}(\mathcal{D}^{(2)}_{s,t})=\Bigg{\\{}(z_{1},z_{2})\in\mathbb{D}^{2}:\sqrt{\frac{2}{t+1}}<\Big{|}\frac{z_{1}-z_{2}}{1-\overline{z_{1}}z_{2}}\Big{|}<\sqrt{\frac{2}{s+1}}\Bigg{\\}}$

and

(2.10) $\displaystyle
J^{-1}(\mathcal{D}^{(2)}_{r})=\Bigg{\\{}(z_{1},z_{2})\in\mathbb{D}^{2}:\Big{|}\frac{z_{1}-z_{2}}{1-\overline{z_{1}}z_{2}}\Big{|}<\sqrt{\frac{2}{r+1}}\Bigg{\\}}.$

So it is natural to introduce domains $\mathbb{D}^{2}_{r}$ and
$\mathbb{D}^{2}_{s,t}$ defined by

$\displaystyle\mathbb{D}^{2}_{r}=\Big{\\{}(z_{1},z_{2})\in\mathbb{D}^{2}:\Big{|}\frac{z_{1}-z_{2}}{1-\overline{z_{1}}z_{2}}\Big{|}<r\Big{\\}},r\in(0,1)$

and

$\displaystyle\mathbb{D}^{2}_{s,t}=\Big{\\{}(z_{1},z_{2})\in\mathbb{D}^{2}:s<\Big{|}\frac{z_{1}-z_{2}}{1-\overline{z_{1}}z_{2}}\Big{|}<t\Big{\\}},0\leq
s<t\leq 1.$

Note that $\mathbb{D}^{2}_{1}=\mathbb{D}^{2}$, if we define
$\mathbb{D}^{2}_{1}$ in a manner similar to the above. Now we shall show that
$Aut(\mathbb{D}^{2}_{r})$ and $Aut(\mathbb{D}^{2}_{s,t})$ can be expressed in
terms of $G_{\mathbb{D}}$. Before going into that result, we want to focus on
a few facts.

  1. (1)

$H^{-1}\mathcal{R}^{(1)}H=G_{\mathbb{D}}$ and $H^{-1}(-I_{3})H=\sigma$, where
$\sigma(z_{1},z_{2})=(z_{2},z_{1})$.
To see this, recall (look at the sentence immediately following (2.6)) that
$\mathcal{R}^{(1)}$ is isomorphic to $SO_{2,1}(\mathbb{R})^{0}$. We know from
[8] that $\mathcal{R}^{(1)}=G(\mathcal{D}^{(2)}_{1,\infty})$ (the latter
denotes the connected identity-component of
$Aut(\mathcal{D}^{(2)}_{1,\infty})$—hereinafter, for $M$ an arbitrary
hyperbolic complex manifold, we shall use $G(M)$ to denote $Aut(M)^{0}$).
Therefore $\mathcal{R}^{(1)}$ is a 3-dimensional Lie group. Since $H$ is a
biholomorphism between $\mathbb{D}^{2}_{0,1}$ and
$\mathcal{D}^{(2)}_{1,\infty}$,
$G(\mathbb{D}^{2}_{0,1})=H^{-1}\mathcal{R}^{(1)}H$ is also a 3-dimensional Lie
group. Recall the subgroup $G_{\mathbb{D}}$ (see (2.1)) of
$\text{Aut}(\mathbb{D}^{2})$. It is clearly a subgroup of
$G(\mathbb{D}^{2}_{0,1})$ and a 3-dimensional Lie group in its own right. But
we already know that $G(\mathbb{D}^{2}_{0,1})$ is 3-dimensional. Therefore,
$G_{\mathbb{D}}$ is a connected 3-dimensional subgroup of the connected
3-dimensional Lie group $H^{-1}\mathcal{R}^{(1)}H$ and, consequently,
coincides with the latter, i.e., $H^{-1}\mathcal{R}^{(1)}H=G_{\mathbb{D}}$ (an
open connected subgroup $K$ of a Lie group $G$ is equal to the connected
component of $G$ containing the identity, i.e., $K=G^{0}$—see, for example,
Lemma 7.12 on page $156$ of [10]). The fact that $H^{-1}(-I_{3})H=\sigma$
follows easily from the definition of $H$.

  2. (2)

$Aut_{CR}(\mathscr{F}_{a})=\\{\Phi,\Phi\circ\sigma:\Phi\in G_{\mathbb{D}}\\}$
for all $a\in(0,1)$.
This is a consequence of the fact that for any $\alpha\in(1,\infty)$,
$Aut_{CR}(\eta^{(2)}_{\alpha})$ is generated by $\mathcal{R}^{(1)}$ and
$-I_{3}$ (see page $669$ of [8]), and
$\mathscr{F}_{a}=H^{-1}(\eta^{(2)}_{\alpha})$ for suitable $a$ and $\alpha$.

  3. (3)

For every holomorphic automorphism $\Phi$ of $\mathbb{D}^{2}_{r}$ (resp.,
$\mathbb{D}^{2}_{s,t}$) and for every $a\in(0,r)$ (resp., $(s,t)$), $\Phi$
maps $\mathscr{F}_{a}$ bijectively to itself and hence
$\Phi|_{\mathscr{F}_{a}}$ is a CR-automorphism of $\mathscr{F}_{a}$.

######  Theorem 2.2.

For $r,s,t$ with $r<1$ and $0\leq s<t\leq 1$, $Aut(\mathbb{D}^{2}_{r})$ and
$Aut(\mathbb{D}^{2}_{s,t})$ are equal to $\\{\Phi,\Phi\circ\sigma:\Phi\in
G_{\mathbb{D}}\\}$, where $\sigma(z_{1},z_{2})=(z_{2},z_{1})$.

###### Proof.

We start with an arbitrary automorphism $\Phi$ of $\mathbb{D}^{2}_{r}$ (resp.,
$\mathbb{D}^{2}_{s,t}$). We fix $a\in(0,r)$ (resp., $(s,t)$) arbitrarily. By
the three observations above, we know that there exists $\varphi\in
Aut(\mathbb{D})$ such that $\Phi|_{\mathscr{F}_{a}}=\Phi_{\varphi}$ or
$\Phi_{\varphi}\circ\sigma$. Now note that $\Phi_{\varphi}$ (or
$\Phi_{\varphi}\circ\sigma$, as the case may be) is itself an automorphism of
$\mathbb{D}^{2}_{r}$ (resp., $\mathbb{D}^{2}_{s,t}$). At any rate, $\Phi$ and
$\Phi_{\varphi}$ (or $\Phi_{\varphi}\circ\sigma$, as the case may be) are both
holomorphic maps on $\mathbb{D}^{2}_{r}$ (resp., $\mathbb{D}^{2}_{s,t}$) that
agree on the subset $\mathscr{F}_{a}$, which is a real 3-dimensional
submanifold of $\mathbb{D}^{2}_{r}$ (resp., $\mathbb{D}^{2}_{s,t}$). We know,
from Corollary 2 of Section 2.6 of Chapter 1 of [4], that if the set of points
of $\mathbb{D}^{2}_{r}$ (resp., $\mathbb{D}^{2}_{s,t}$) at which two
holomorphic functions defined thereon agree (which is necessarily an analytic
subset of $\mathbb{D}^{2}_{r}$ (resp., $\mathbb{D}^{2}_{s,t}$)) has Hausdorff
dimension greater than $2\times 2-2=2$, then the two functions are identically
equal. In our case, both the first and second component functions of $\Phi$
and $\Phi_{\varphi}$ (or $\Phi_{\varphi}\circ\sigma$, as the case may be)
agree on the subset $\mathscr{F}_{a}$ of $\mathbb{D}^{2}_{r}$ (resp.,
$\mathbb{D}^{2}_{s,t}$), which has Hausdorff dimension 3, being a real
3-dimensional submanifold of $\mathbb{D}^{2}_{r}$ (resp.,
$\mathbb{D}^{2}_{s,t}$). Therefore, both the pairs are identically equal to
one another, and this implies that $\Phi=\Phi_{\varphi}$ (or
$\Phi_{\varphi}\circ\sigma$, as the case may be). This completes the proof.

∎

Remark: Another proof of Theorem 2.2 can be given using the properties of the
symmetrized bidisc. We shall give a sketch of the proof.

  1. (1)

It suffices to show that if $\Phi\in Aut(\mathbb{D}^{2}_{s,t})$ (or
$Aut(\mathbb{D}^{2}_{r})$) and $\Phi|_{\mathscr{F}_{a}}=id$ for some $a>0$,
then $\Phi=id$. (The reason why this suffices is that, as we know, if we fix
$a\in(s,t)$ (resp., $(0,r)$) arbitrarily, then there exists $\varphi\in
Aut(\mathbb{D})$ such that $\Phi|_{\mathscr{F}_{a}}=\Phi_{\varphi}$ or
$\Phi_{\varphi}\circ\sigma$. Therefore we can consider the automorphism
$\Phi_{\varphi}\circ\Phi^{-1}$ (or
$(\Phi_{\varphi}\circ\sigma)\circ\Phi^{-1}$, as the case may be) and attempt
to conclude that it is equal to the identity.)

  2. (2)

Define $sym:\mathbb{D}^{2}\rightarrow\mathbb{G}$ by
$sym(z_{1},z_{2})=(z_{1}+z_{2},z_{1}z_{2})$,
$\mathbb{G}_{r}=sym(\mathbb{D}^{2}_{r})$ and
$\mathbb{G}_{s,t}=sym(\mathbb{D}^{2}_{s,t})$. Using the map in Theorem $3.2$
in [2] and properties stated in [8] (page 654–655), we see that
$Aut(\mathbb{G}_{r})=Aut(\mathbb{G}_{s,t})=\\{H_{\varphi}:\varphi\in
Aut(\mathbb{D})\\}$ where $H_{\varphi}\circ
sym(z_{1},z_{2})=sym(\varphi(z_{1}),\varphi(z_{2}))$.

  3. (3)

Define $\Psi_{\Phi}:\mathbb{G}_{s,t}\rightarrow\mathbb{C}^{2}$ by
$\Psi_{\Phi}\circ sym(z_{1},z_{2})=sym\circ\Phi(z_{1},z_{2})$. It is easy to
see that $\Psi_{\Phi}$ is well-defined and is a holomorphic automorphism of
$\mathbb{G}_{s,t}$ (respectively, of $\mathbb{G}_{r}$, when defined
similarly). So $\Psi_{\Phi}=H_{\varphi}$ for some $\varphi\in
Aut(\mathbb{D})$. Using $\Phi|_{\mathscr{F}_{a}}=id$, we get $\varphi=id\in
Aut(\mathbb{D})$.

  4. (4)

So for any $(z_{1},z_{2})$, $\Phi(z_{1},z_{2})$ is either $(z_{1},z_{2})$ or
$(z_{2},z_{1})$. A simple argument involving sequences shows that $\Phi=id$ in
a neighbourhood of $(a,0)$ and this will imply the conclusion of Theorem 2.2.

In our article, we shall use the domains $\mathbb{D}^{2}_{r}$ to prove our
main result. By [8], it is a hyperbolic 2-manifold with 3-dimensional
automorphism group. Our next result is a characterization of
$\mathbb{D}^{2}_{r}$.

######  Theorem 2.3.

Suppose $M$ is a hyperbolic 2-manifold with 3-dimensional automorphism group.
Let $G(M):=Aut(M)^{0}$. Suppose that $M$ has an orbit under the action of
$G(M)$ that is a complex curve, that there exist an $a\in(0,1)$ and a
3-dimensional orbit $O$ of $M$ under $G(M)$ that is CR-equivalent to
$\mathscr{F}_{a}$ and that all remaining orbits are strongly pseudoconvex real
hypersurfaces. Then there exists an $r\in(0,1)$ such that $M$ is biholomorphic
to $\mathbb{D}^{2}_{r}$.

###### Proof.

Our theorem follows, with very little effort, from Isaev’s work. It follows
from the proof of [8, Theorem 5.1] that if $M$ is a hyperbolic 2-manifold with
3-dimensional automorphism group having an orbit under the action of $G(M)$
that is a complex curve, such that all its remaining orbits are strongly
pseudoconvex (3-dimensional) real hypersurfaces and such that one of these
3-dimensional orbits is CR-equivalent to $\eta^{(2)}_{\alpha}$ for some
$\alpha\in(1,\infty)$, then $M$ is biholomorphic to $\mathcal{D}^{(2)}_{s}$
for some $s\in(1,\infty)$. But we have already seen that $\eta^{(2)}_{\alpha}$
is CR-equivalent to $\mathscr{F}_{a}$, where

$\alpha=\frac{8}{a^{4}}-\frac{8}{a^{2}}+1,$

and $\mathcal{D}^{(2)}_{s}$ is biholomorphic with some $\mathbb{D}^{2}_{r}$.
Therefore, we can use the observation that we stated above (see (2.10)) to
conclude that $M$ is biholomorphic to $\mathbb{D}^{2}_{r}$ for some $r<1$. ∎

Now suppose that, with $M$ as above, we have the following:

  1. (1)

Exactly one $G(M)$-orbit is a complex curve.

  2. (2)

The remaining orbits are strongly pseudoconvex real hypersurfaces.

  3. (3)

There exist a sequence $\\{a_{n}\\}\subset(0,1)$ converging to $1$, and, for
each $n$, an orbit $\mathcal{O}_{n}$ in $M$ which is CR-equivalent to
$\mathscr{F}_{a_{n}}$.

Under these conditions, $M$ cannot be biholomorphic with $\mathbb{D}^{2}_{r}$
for any $r\in(0,1)$. The reason behind this claim is the following: If
$b_{1},b_{2}\in(0,1)$ are two distinct numbers, then $\mathscr{F}_{b_{1}}$ and
$\mathscr{F}_{b_{2}}$ are not CR-equivalent (they are CR-equivalent to
$\eta^{(2)}_{\beta_{1}}$ and $\eta^{(2)}_{\beta_{2}}$, respectively, where

$\beta_{j}=\frac{8}{b_{j}^{4}}-\frac{8}{b_{j}^{2}}+1,\;j=1,2;$

hence $\beta_{1}\neq\beta_{2}$ and so, by the remark made immediately after
(2.7), $\mathscr{F}_{b_{1}}$ and $\mathscr{F}_{b_{2}}$ are CR-nonequivalent.)
If $M$ is biholomorphic with $\mathbb{D}^{2}_{r}$ for some $r\in(0,1)$, then,
considering the action of $G_{\mathbb{D}}$ on $\mathbb{D}^{2}_{r}$ and the
orbits of $\mathbb{D}^{2}_{r}$ under $G_{\mathbb{D}}$, we are led to conclude
that there exists an orbit of $\mathbb{D}^{2}_{r}$ under $G_{\mathbb{D}}$ that
is CR-equivalent to some $\mathscr{F}_{a_{n}}$, where $a_{n}\in(r,1)$. But,
taking into consideration the CR-nonequivalence of the $\mathscr{F}_{a}$’s,
this is impossible.

###  2.2. Action of $SU_{1,1}(\mathbb{C})$ and $O_{2,1}(\mathbb{R})$ on the
open unit ball

Let $\mathbb{B}_{2}$ denote the open Euclidean unit ball in $\mathbb{C}^{2}$.
Its automorphism group has a connection with the Lie group
$SU_{2,1}(\mathbb{C})$, which acts on $\mathbb{B}_{2}$ in the following way

$\begin{pmatrix}a_{1}&a_{2}&a_{3}\\\ b_{1}&b_{2}&b_{3}\\\
c_{1}&c_{2}&c_{3}\end{pmatrix}\cdot(u,v)=\frac{(a_{1}u+a_{2}v+a_{3},b_{1}u+b_{2}v+b_{3})}{c_{1}u+c_{2}v+c_{3}}.$

In this subsection we consider $O_{2,1}(\mathbb{R})$ and a copy of
$SU_{1,1}(\mathbb{C})$ as subgroups of $SU_{2,1}(\mathbb{C})$ and study their
actions on $\mathbb{B}_{2}$.

First we investigate the action of $SU_{1,1}(\mathbb{C})$. Consider

$\begin{pmatrix}\alpha&\beta\\\
\overline{\beta}&\overline{\alpha}\end{pmatrix}\mapsto\begin{pmatrix}1&0&0\\\
0&\alpha&\beta\\\ 0&\overline{\beta}&\overline{\alpha}\end{pmatrix}$

where $|\alpha|^{2}-|\beta|^{2}=1$. This is an embedding of
$SU_{1,1}(\mathbb{C})$ in $SU_{2,1}(\mathbb{C})$. We shall identify
$SU_{1,1}(\mathbb{C})$ with the subgroup

$\displaystyle\Bigg{\\{}\begin{pmatrix}1&0&0\\\ 0&\alpha&\beta\\\
0&\overline{\beta}&\overline{\alpha}\end{pmatrix}:\alpha,\beta\in\mathbb{C},|\alpha|^{2}-|\beta|^{2}=1\Bigg{\\}}$

of $SU_{2,1}(\mathbb{C})$ and observe its action on $\mathbb{B}_{2}$.

######  Proposition 2.4.

The $SU_{1,1}(\mathbb{C})$-orbits of $\mathbb{B}_{2}$ are given by
$SU_{1,1}(\mathbb{C})\cdot(t,0),\,\,t\in[0,1)$.
$SU_{1,1}(\mathbb{C})\cdot(0,0)$ is a complex curve and, for $t\in(0,1)$,
$SU_{1,1}(\mathbb{C})\cdot(t,0)$ is CR-equivalent with the sphere, that is,
the topological boundary of $\mathbb{B}_{2}$.

###### Proof.

Let $(u,v)\in\mathbb{B}_{2}$. For

$A=\begin{pmatrix}1&0&0\\\ 0&\alpha&\beta\\\
0&\overline{\beta}&\overline{\alpha}\end{pmatrix},\,\,\text{we
have}\,\,A\cdot(u,v)=\frac{(u,\alpha
v+\beta)}{\overline{\beta}v+\overline{\alpha}}.$

Note that $v\in\mathbb{D}$ and $\frac{\alpha
v+\beta}{\overline{\beta}v+\overline{\alpha}}$ is the action of
$\begin{pmatrix}\alpha&\beta\\\
\overline{\beta}&\overline{\alpha}\end{pmatrix}$ on $v$. Since the action of
$SU_{1,1}(\mathbb{C})$ (considered as $2\times 2$ matrices) is transitive on
$\mathbb{D}$, a little calculation gives that with proper choice of $A$, we
can find a $t\in[0,1)$ such that $A\cdot(u,v)=(t,0)$. If for a fixed $(u,v)$,
there are two elements of $(0,1)$, say $t_{1},t_{2}$, such that $(t_{1},0)$
and $(t_{2},0)$ both belong to the $SU_{1,1}(\mathbb{C})$-orbit of $(u,v)$,
then that means that there exists an element of $SU_{1,1}(\mathbb{C})$
carrying $(t_{1},0)$ to $(t_{2},0)$, hence, from the above, that there exist
complex numbers $\alpha,\beta$ such that
$\frac{(t_{1},\beta)}{\overline{\alpha}}=(t_{2},0)$ with
$|\alpha|^{2}-|\beta|^{2}=1$. This gives us that $\beta=0$. If $t_{1}=0$, then
$t_{2}=0$. When at least one of $t_{1},t_{2}$ is positive, the other one is
also positive, we get $\alpha=1$ and we also get $t_{1}=t_{2}$. Thus the $t$
above is unique and consequently the $SU_{1,1}(\mathbb{C})$-orbits of
$\mathbb{B}_{2}$ are $SU_{1,1}(\mathbb{C})\cdot(t,0),\,\,t\in[0,1)$.

Now consider
$SU_{1,1}(\mathbb{C})\cdot(0,0)=\\{(0,\frac{\beta}{\overline{\alpha}}):\alpha,\beta\in\mathbb{C},|\alpha|^{2}-|\beta|^{2}=1\\}$.
With a little effort we can show that the map
$f:\mathbb{D}\rightarrow\mathbb{C}^{2}$ sending $z$ to $(0,z)$ maps
$\mathbb{D}$ biholomorphically onto $SU_{1,1}(\mathbb{C})\cdot(0,0)$.

Focusing on the remaining orbits, we find that for $t\in(0,1)$,
$SU_{1,1}(\mathbb{C})\cdot(t,0)=\\{(u,v)\in\mathbb{B}_{2}:|u|^{2}+t^{2}|v|^{2}=t^{2}\\}$.
Now it is easy to see that the biholomorphic map
$g_{t}:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ sending $(z,w)$ to
$(\frac{z}{t},w)$ maps $SU_{1,1}(\mathbb{C})\cdot(t,0)$ onto $S^{3}$, the
topological boundary of $\mathbb{B}_{2}$. This completes the proof. ∎

Now let us focus on the action of $O_{2,1}(\mathbb{R})$. By
$O_{2,1}(\mathbb{R})$, we mean the subgroup $\\{A\in SU_{2,1}(\mathbb{C}):A\in
M_{3}(\mathbb{R}),A^{t}I_{2,1}A=I_{2,1}\\}$ of $SU_{2,1}(\mathbb{C})$. We only
take a look at the $O_{2,1}(\mathbb{R})$-orbit of $(0,0)$.

######  Proposition 2.5.

The $O_{2,1}(\mathbb{R})$-orbit of $(0,0)$ is totally real.

###### Proof.

If $A=\begin{pmatrix}a_{1}&a_{2}&a_{3}\\\ b_{1}&b_{2}&b_{3}\\\
c_{1}&c_{2}&c_{3}\end{pmatrix}\in O_{2,1}(\mathbb{R})$, then
$A\cdot(0,0)=(\frac{a_{3}}{c_{3}},\frac{b_{3}}{c_{3}})$. So
$O_{2,1}(\mathbb{R})\cdot(0,0)\subset\\{(u,v)\in\mathbb{B}_{2}:Im(u)=0=Im(v)\\}$.
Suppose $(z,w)\in\mathbb{B}_{2}$ with $Im(z)=0=Im(w)$. If $(z,w)=(0,0)$, then
we have $(z,w)=I_{3}\cdot(0,0)$. If $(z,w)\neq(0,0)$, then consider the matrix

$\displaystyle B=\begin{pmatrix}-\alpha w&kz&\gamma z\\\ \alpha z&kw&\gamma
w\\\ 0&k(z^{2}+w^{2})&\gamma\end{pmatrix}$

where
$\alpha=\frac{1}{\sqrt{z^{2}+w^{2}}},k=\frac{1}{\sqrt{(z^{2}+w^{2})(1-z^{2}-w^{2})}}$
and $\gamma=\frac{1}{\sqrt{1-z^{2}-w^{2}}}$. It is easy to see that $B\in
O_{2,1}(\mathbb{R})$ and $B\cdot(0,0)=(z,w)$. Hence we get that
$O_{2,1}(\mathbb{R})\cdot(0,0)=\\{(u,v)\in\mathbb{B}_{2}:Im(u)=0=Im(v)\\}$.
This proves the proposition. ∎

##  3\. Main result

Before stating our main result, we want to remind the reader that given a
2-dimensional hyperbolic manifold $M$ with a 3-dimensional subgroup $G$ of
$Aut(M)^{0}$, there are only a few possibilities for
$dim_{\mathbb{R}}Aut(M)^{0}$. Obviously, $dim_{\mathbb{R}}Aut(M)^{0}\geq 3$.
It is a standard result (see, for example, Section 1.2 of [9] and, in
particular, equation (1.3) therein) that for an $n$-dimensional hyperbolic
complex manifold $M$, $dim_{\mathbb{R}}Aut(M)\leq n^{2}+2n$. In our case, this
gives us: $dim_{\mathbb{R}}Aut(M)\leq 8$. Furthermore, we may conclude, using
Corollary 2.5 of [9], that $dim_{\mathbb{R}}Aut(M)\neq 5$ (see also Theorem
2.2 therein). Finally, we may conclude, using Theorem 1.3 of [9], that
$dim_{\mathbb{R}}Aut(M)\neq 7$. This shows that $dim_{\mathbb{R}}Aut(M)$, or,
equivalently, $dim_{\mathbb{R}}Aut(M)^{0}$ is one of $3,4,6$ and $8$. When
$dim_{\mathbb{R}}Aut(M)^{0}=8$, $M$ is biholomorphic with $\mathbb{B}_{2}$
(see, for example, Theorem 1.1 of [9]) and when
$dim_{\mathbb{R}}Aut(M)^{0}=6$, $M$ is biholomorphic with $\mathbb{D}^{2}$
(see, for example, Corollary 2.5 of [9]). The cases of $3$ and $4$ are more
complicated and can be found in [7] and [8]. We shall show that under certain
conditions, found by observing the action of $G_{\mathbb{D}}$ on
$\mathbb{D}^{2}$, the only possibility is $dim_{\mathbb{R}}Aut(M)^{0}=6$.

######  Theorem 3.1.

Let $M$ be a hyperbolic 2-manifold with a 3-dimensional non-solvable subgroup
$G$ of $G(M)=Aut(M)^{0}$. Suppose that, under the action of $G$, the following
conditions are met.

  1. (1)

Exactly one $G$-orbit is a complex curve.

  2. (2)

The remaining $G$-orbits are strongly pseudoconvex real hypersurfaces.

  3. (3)

There exist a sequence $\\{a_{n}\\}\subset(0,1)$ converging to $1$ and, for
each $n$, a $G$-orbit $\mathcal{O}_{n}$ in $M$ which is CR-equivalent to
$\mathscr{F}_{a_{n}}$.

Then $M$ is biholomorphic with the bidisc $\mathbb{D}^{2}$.

###### Proof.

All we have to do is to discard the cases $dim_{\mathbb{R}}G(M)=3,4,8$. Once
this is done, it will follow that $M$ is biholomorphic with $\mathbb{D}^{2}$,
because, as mentioned above, it is known that the bidisc is, up to
biholomorphic equivalence, the unique 2-dimensional hyperbolic complex
manifold with 6-dimensional automorphism group.

Case (i): $dim_{\mathbb{R}}G(M)\neq 3$.
Assume, to get a contradiction, that $dim_{\mathbb{R}}G(M)=3$. Then using
Theorem 2.3, we conclude that $M$ is biholomorphic with $\mathbb{D}^{2}_{r}$
for some $r\in(0,1)$. But the discussion following the proof of Theorem 2.3
forbids this possibility. So $dim_{\mathbb{R}}G(M)\neq 3$.

Case (ii): $dim_{\mathbb{R}}G(M)\neq 4$.
Assume, to get a contradiction, that $dim_{\mathbb{R}}G(M)=4$. Then $M$ cannot
be homogeneous, otherwise by Theorem $1.1$ in [7], $M$ must be of dimension
$3$ or $4$. By Proposition 2.1. in [7], each $G(M)$-orbit is either a real
hypersurface or a complex curve. Clearly our manifold $M$ contains at most one
complex curve and some real 3-dimensional hypersurfaces as $G(M)$-orbits. The
$G(M)$-orbits have a connection with the $G$-orbits. If $O_{G(M)}(p)$ and
$O_{G}(p)$ denote the $G(M)$-orbit and the $G$-orbit of $p\in M$,
respectively, then for any $q\in O_{G(M)}(p)$, we must have $O_{G}(q)\subset
O_{G(M)}(p)$. Also $q\in O_{G}(q)$ for all $q\in M$. Consequently
$O_{G(M)}(p)=\bigcup_{q\in O_{G(M)}(p)}O_{G}(q)$ for all $p\in M$. This
implies, in particular, that there is a 3-dimensional $G(M)$-orbit
$\mathcal{O}$ that contains a $G$-orbit $\widetilde{\mathcal{O}}$ CR-
equivalent to some $\mathscr{F}_{a}$. Since both $\mathcal{O}$ and
$\widetilde{\mathcal{O}}$ are 3-dimensional, it follows that
$\widetilde{\mathcal{O}}$ is an open subset of $\mathcal{O}$. Now by
Proposition 2.1. in [7], $\mathcal{O}$ is either Levi flat or spherical while
each $\mathscr{F}_{a_{m}}$ is a 3-dimensional hypersurface which is strongly
pseudoconvex and non-spherical (recall, from Proposition 2.1, that
$\mathscr{F}_{a_{m}}$ is CR-isomorphic to $\eta^{(2)}_{\alpha}$ for some
$\alpha$, and $\eta^{(2)}_{\alpha}$ is strongly pseudoconvex and non-
spherical—see [3], [6], [8]). In particular, $\mathcal{O}$, which is either
Levi-flat or spherical, has a non-empty open subset,
$\widetilde{\mathcal{O}}$, which is strongly pseudoconvex and non-spherical.
This is a contradiction. Hence $dim_{\mathbb{R}}G(M)\neq 4$.

Case (iii): $dim_{\mathbb{R}}G(M)\neq 8$.
Assume, to get a contradiction, that $\dim_{\mathbb{R}}G(M)=8$. Then, by
Theorem 1.1 of [9], $M$ is biholomorphic to $\mathbb{B}_{2}$. Therefore it
follows from the hypotheses that $G(\mathbb{B}_{2})$ has a certain non-
solvable 3-dimensional Lie subgroup $G$ that satisfies the following
properties: (1) the orbits of $\mathbb{B}_{2}$ under $G$, except only one, are
strongly pseudoconvex 3-dimensional real hypersurfaces and the sole remaining
orbit is a complex curve, and (2) there exist an $a\in(0,1)$ and a
3-dimensional orbit of $\mathbb{B}_{2}$ under $G$ that is CR-equivalent to
$\mathscr{F}_{a}$. Now the automorphism group of $\mathbb{B}_{2}$ is known
(see Proposition 3, Section 7, Chapter 2 of [1]) to be

$PSU_{2,1}(\mathbb{C})=SU_{2,1}(\mathbb{C})/K,$

where $K=\\{\lambda I_{3}\mid\lambda^{3}=1\\}$. The Lie algebra of $G$ is, by
our assumption, non-solvable. Now the conjugacy classes of all the Lie
subgroups of $SU_{2,1}(\mathbb{C})$ have been determined in [11] and listed in
Table II on page 1390 of [11]. Our group $G$ is (assumed to be isomorphic to)
a Lie subgroup of $SU_{2,1}(\mathbb{C})/K$ and must, therefore, be of the form
$N/K$, where $N$ is some Lie subgroup of $SU_{2,1}(\mathbb{C})$. Thus $N$ can
be identified with a group of matrices that is a subgroup of
$SU_{2,1}(\mathbb{C})$. Note that $SU_{2,1}(\mathbb{C})$ itself acts on
$\mathbb{B}_{2}$, as we have discussed in subsection 2.2. Note further that
$N$ is a subgroup of $SU_{2,1}(\mathbb{C})$ containing $K$, and that the
orbits of $\mathbb{B}_{2}$ under $N$ are precisely the orbits of
$\mathbb{B}_{2}$ under $N/K$ (this is just a reflection of the general fact
that if we take any two matrices $A$ and $B$ in $SU_{2,1}(\mathbb{C})$ such
that there exists $\lambda\in\mathbb{C}$ such that $B=\lambda A$, then
($\lambda^{3}=1$) and for every $\underline{z}\in\mathbb{B}_{2}$,
$A\cdot\underline{z}=B\cdot\underline{z}$, where $\cdot$ denotes the action of
$SU_{2,1}(\mathbb{C})$ on $\mathbb{B}_{2}$ that was discussed in subsection
2.2). Now $N$ must belong to some conjugacy class appearing in the
aforementioned table. Since $G$ is isomorphic to $N/K$, and $G$ is assumed to
be non-solvable, $N$ must be non-solvable as well. Therefore, from the table,
we see that $N$ must belong either to the conjugacy class of
$SU_{1,1}(\mathbb{C})$ or to that of $SU_{2}(\mathbb{C})$ or to that of
$O_{2,1}(\mathbb{R})$ (that is, isomorphic to some embedded copy as we have
discussed in subsection 2.2). It cannot belong to the conjugacy class of
$SU_{2}(\mathbb{C})$, because, if it did, then $G$ would be compact, which
cannot be the case (by assumption, it acts on $\mathbb{B}_{2}$ to produce non-
compact orbits — $\mathscr{F}_{a}$ is non-compact). Consequently, $N$ must
belong either to the conjugacy class of $SU_{1,1}(\mathbb{C})$ or to that of
$O_{2,1}(\mathbb{R})$.

Suppose first that $N$ belongs to the conjugacy class of
$SU_{1,1}(\mathbb{C})$. Consider the orbits of $\mathbb{B}_{2}$ under $N$. It
has been shown that the 3-dimensional orbits of $\mathbb{B}_{2}$ under
$SU_{1,1}(\mathbb{C})$ are spherical hypersurfaces (Proposition 2.4). Our
assumption implies that the orbits of $\mathbb{B}_{2}$ under $N/K$ are, with
one exception, strongly pseudoconvex hypersurfaces, that the sole remaining
orbit is a complex curve, and that there exist $a\in(0,1)$ and a hypersurface
orbit that is CR-equivalent to $\mathscr{F}_{a}$. By the observation about the
equality of the orbits of $\mathbb{B}_{2}$ under $N$ and $N/K$, it follows
that there exists an orbit $\mathcal{O}$ of $\mathbb{B}_{2}$ under $N$ that is
CR-equivalent to $\mathscr{F}_{a}$ for some $a\in(0,1)$. Now, since $N$
belongs to the conjugacy class of $SU_{1,1}(\mathbb{C})$, it follows that
$\mathcal{O}$ is the image of a spherical hypersurface in $\mathbb{B}_{2}$
under a holomorphic automorphism of $\mathbb{B}_{2}$ and hence is spherical
itself. But that is a contradiction, because $\mathcal{O}$ is CR-equivalent to
$\mathscr{F}_{a}$, which is not spherical.

Suppose now that $N$ belongs to the conjugacy class of $O_{2,1}(\mathbb{R})$.
It has been shown that the orbit of $(0,0)$ under $O_{2,1}(\mathbb{R})$ is a
totally real 2-dimensional submanifold of $\mathbb{B}_{2}$ (Proposition 2.5).
Consequently, among the orbits of $\mathbb{B}_{2}$ under $N$, there must exist
a totally real 2-dimensional submanifold. But there does not, because, as
above, it follows from our assumption about the orbits of $\mathbb{B}_{2}$
under $N/K$ that the only 2-dimensional orbit of $\mathbb{B}_{2}$ under $N$ is
a complex curve.

Therefore, we invariably get a contradiction. This shows that
$\dim_{\mathbb{R}}G(M)\neq 8$.

Thus, $dim_{\mathbb{R}}G(M)=6$, and this completes the proof.

∎

Acknowledgements: The first-named author’s research is supported by the
University Grants Commission Centre for Advanced Studies. The second-named
author’s research is supported by a postdoctoral fellowship from the Harish-
Chandra Research Institute (Homi Bhabha National Institute).

## References

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# Petroleum prices prediction using data mining techniques – A Review

John Ngechu  dept.Computer Science
Dedan Kimathi University of Technology
<EMAIL_ADDRESS>Kiplang’at Weldon  dept. Computer Science
Dedan Kimathi University of Technology
<EMAIL_ADDRESS>Ngatho Everlyne  dept. Computer Science
Dedan Kimathi University of Technology
<EMAIL_ADDRESS>Nancy Njambi  dept. Computer Science
Dedan Kimathi University of Technology
<EMAIL_ADDRESS>Kinyua Gikunda  dept. Computer Science
Dedan Kimathi University of Technology
<EMAIL_ADDRESS>
###### Abstract

Over the past 20 years, Kenya’s demand for petroleum products has
proliferated. This is mainly because this particular commodity is used in many
sectors of the country’s economy. Exchange rates are impacted by constantly
shifting prices, which also impact Kenya’s industrial output of commodities.
The cost of other items produced and even the expansion of the economy is
significantly impacted by any change in the price of petroleum products.
Therefore, accurate petroleum price forecasting is critical for devising
policies that are suitable to curb fuel-related shocks. Data mining techniques
are the tools used to find valuable patterns in data. Data mining techniques
used in petroleum price prediction, including artificial neural networks
(ANNs), support vector machines (SVMs), and intelligent optimization
techniques like the genetic algorithm (GA), have grown increasingly popular.
This study provides a comprehensive review of the existing data mining
techniques for making predictions on petroleum prices. The data mining
techniques are classified into regression models, deep neural network models,
fuzzy sets and logic, and hybrid models. A detailed discussion of how these
models are developed and the accuracy of the models is provided.

###### Index Terms:

Data mining techniques, petroleum prices, prediction.

##  I Introduction

The most important source of energy for economic activity is petroleum, which
is also the most traded commodity on the globe[1]. Petroleum is essential to
industry and civilization, and as it supplies a sizable amount of the world’s
energy needs, it plays a significant role in world politics and international
relations[2]. With 33% of the primary energy consumed globally, it continues
to be the most significant energy source in the world. Its dominance in the
transportation industry, which accounts for 94% of all energy used, highlights
its importance[3].The expense of fuel has long caused concern across all
social groups. Fluctuations in the price of petroleum have a direct impact on
the economy of the nation, impacting retailers, consumers, the production
industries, the public and private transportation sectors. Additionally, it
has an indirect effect on the price of commodities, the stock market,
inflation, and other significant aspects of modern man’s life cycle[4].

Numerous variables affect the price of petroleum. First, just as with any
other commodity, economic factors such as petroleum supply and demand greatly
influence the rise and fall of petroleum prices[5]. Second, the price of
petroleum is influenced by the chemical characteristics of oil, such as
density and viscosity, as well as by technical elements of the oil sector,
such as refining capacity and above-ground oil stocks. Additionally,
production techniques like shale oil extraction have a substantial influence
on the supply of crude oil and, consequently, the price of petroleum[6].
Third, geopolitical conflicts and tensions have a big impact on oil prices.
Because the Middle East has the largest oil reserves in the world and is a
politically unstable region, geopolitical factors like the Russia-Ukraine war
have had a big influence on oil prices[7]. By impacting the supply of
petroleum, economic sanctions imposed on significant oil-producing nations can
also cause disruptions in the world energy markets.

To shed light on the behavior of oil prices, several attempts have been made.
Two key categories are taken into consideration while estimating the price of
petroleum. The first group includes conventional statistical and econometric
approaches.This includes the exponential smoothing model (ESM), linear
regression (LinR), autoregressive integrated moving average (ARIMA),
generalized autoregressive conditional heteroskedasticity (GARCH), random walk
(RW), and error correction models (ECM)[8]. Only linear trends may often be
detected in time-series data using traditional statistical and econometric
tools. These models may fall short of capturing the nonlinear characteristics
of crude oil pricing[9]. To get around this constraint, Data Mining techniques
using AI models[10]with strong self-learning abilities are used.

Data mining is the process of discovering patterns and trends to extract
valuable information from large data collections to assess or make a
decision[11]. In order to find the valuable patterns in data, data mining
techniques are the tools used[12]. Data mining techniques used in petroleum
price prediction includes, artificial neural networks (ANNs), support vector
machines (SVMs), and intelligent optimization techniques like the genetic
algorithm (GA) have grown increasingly popular[13].

Due to a multitude of natural factors, petroleum prices are very erratic, vary
more than other mineral resources, and are challenging to predict[3]. On the
one hand, accurate forecasting is critical for private investors and central
banks, who rely on accurate oil price forecasts to devise suitable policies in
reaction to petroleum-related shocks. As a result, a wide spectrum of
organizations seeks precise oil price forecasts[10].

Since petroleum is the most major source of energy and is used in practically
every area of our life, it has a big impact on how our economy develops.As a
result, shifts in the price of petroleum are frequently excellent indications
of shifts in the national economy and worldwide markets[14]. The ability to
anticipate these swings in petroleum prices helps economic players, such as
companies, to adapt to impending market changes and provides decision-makers
with accurate knowledge with which they may use it to decide on the best
course of action[15].

The following are some reasons for doing the research: a) Data mining
techniques help quickly automate predictions of behaviors and trends and
discover hidden patterns. (b) Data mining has been used in many domains for
price prediction. (c) Data mining helps ease the process of analyzing enormous
amounts of data. (d) Data mining helps in making informed decisions. This
study aims to discover petroleum price prediction research and development and
provide insight into the implementation of data mining techniques in relation
to petroleum price prediction. As a result, the investigation’s main focus
will be on how well data mining tools can forecast petroleum prices in order
to inform future researchers. In this light, the study’s research questions
are: a) What is the role of data mining in petroleum price prediction? b) What
type of data mining techniques should be used? c) What are the benefits of
using data mining techniques in petroleum price prediction in Kenya?

##  II METHODOLOGY

In order to address the research questions, a bibliographic analysis of the
subject under study was carried out between 2019 and 2022. There were two
steps involved: (a) clustering of related works and (b) detailed review and
analysis of the works. In the first stage, a keyword-based search is
conducted, utilizing all possible combinations of two sets of keywords, the
first of which is concerned with data mining techniques and the second group
refers to petroleum price prediction (i.e., crude oil price, oil price, fuel
price). The analysis was done while considering the following research
questions: (a) petroleum product predicted, (b) data set used, (c) based on
the author’s performance metric, and accuracy, and (d) data mining techniques
used.

##  III AN OVERVIEW OF DATA MINING TECHNIQUES

Data mining is the process of detecting or extracting fascinating patterns,
correlations, changes, anomalies, and key structures from enormous amounts of
data held in various data sources such as file systems, databases, data
warehouses, and other information repositories[16]. As a multidisciplinary
field, data mining utilizes a number of techniques or approaches from various
disciplines, including statistics, machine learning, neural networks, genetic
algorithms, fuzzy sets, and visualization [17].

###  III-A Single Statistical and Econometric Prediction Model

Regression models are the fundamental time series forecasting methods. The
ARIMA equation, a linear equation, is the most often used regression
model[18]. ARIMA model has been widely used to predict fuel prices because of
its ease of use[19]. The ARIMA (p,d,q) standard model is described as:

$\varphi_{p}(B)\nabla^{d}y_{t}=\mu+\theta_{q}(B)\varepsilon_{t}$ (1)

[21] applied the autoregressive integrated moving average model to predict
Fuel prices in Malaysia. The data source that was used for that particular
study was the weekly price of Ron97 which was made public by the ministry of
trade in Malaysia. ARIMA model used to forecast petroleum fuel prices in
Malaysia has been proven to predict the fuel prices accurately for shorter
periods as compared to the longer periods. A study was conducted by [22] to
improve the predictions of fuel cost distribution using the ARIMA model.The
data for this study was obtained from the US Energy Information Administration
(EIA)-923 from January 2013 to December 2016. Data selection is done first in
order to explore the data and come up with a fuel cost time series. The fuel
price time series is in turn used in the ARIMA model the fuel price for the
next month. To validate the model, 6 months are used from July 2016 to
December 2016. A normal distribution is then used to fit the data to obtain an
estimated fuel price. Kullback-Leibler divergence is then used to evaluate the
performance of the algorithm [23]. The results from the predictions indicate
that the model have higher accuracy as compared to using three-month delayed
EIA data when the market does not have high volatility. [24] proposed a model
to predict petroleum prices in Kenya using the Autoregressive integrated
moving average (ARIMA) and the Vector Autoregressive models(VAR). The data
sources from the research were crude oil prices in the world market, exchange
rates, inflation rates and the secondary petroleum pump prices data from the
Energy and Petroleum Regulatory Authority of Kenya. The data used was from
January 2011 to December 2018. After developing the model’s prediction was
made for the next twelve months. The petroleum prices were discovered to be
non-stationery. The prediction accuracy of the two models ARIMA and VAR was
compared using root mean square error (RMSE), mean absolute error (MAE) and
mean absolute percentage error (MAPE).

TABLE I: Comparison of ARIMA and VAR model[24].

From the study, VAR was better in the prediction of the petroleum prices in
Kenya as compared to the ARIMA model.

###  III-B Deep Neural Networks.

Deep learning is a subset of machine learning techniques that is based on
learning data representations[25].Data extraction and manipulation may be
carried out automatically using neurons, a network of multi-layered non-linear
processing units utilized in deep learning models. An artificial neural
network is a network made up of these neurons. Node clusters in ANN layers are
joined to produce a picture of the human brain. The input for the subsequent
layer is the yield of the previous layer. These system models typically pick
up new information from their training data [26]

  1. (a)

CNN: A novel crude oil price forecasting approach based on online media text
mining to capture the more immediate market antecedents of price swings[27].
Based on the CNN model, sentiment analysis, and topic identification (Latent
Dirichlet Allocation (LDA) topic model), a text-based topic-sentiment
synthesis technique was presented for time series generation. By taking into
account unanticipated political or social developments that are reported in
the press, this strategy could improve crude oil price prediction accuracy.
The CNN model’s accuracy was 61 percent, which was lower than projected.
Rolling test windows for each and every news headline, like in normal time
series forecasting, yielded a lower accuracy of 59.8

Selvin, S. used CNN, LSTM, and RNN architectures to forecast the price of NSE-
listed companies and compared their performance. The final results showed that
CNN is the best architecture to predict the price of stock because it could
identify the trend of the directional change [28]. Liu, S. proposed a CNN-LSTM
model, and the model performed a basic momentum strategy and benchmark model
for which the return rates were 0.882 and 1.136, respectively. The CNN part
could extract useful features even from low signal-to-noise time-series data,
and the LSTM part could predict future stock prices with high accuracy. Then,
the predicting outcomes were used as timing signals [29].

  2. (b)

LSTM: Long Short-Term Memory (LSTM), also known as an improved recurrent
neural network (RNN) model. In RNN, the output from the previous step is
provided as input during the current phase. It addressed the issue of long-
term dependency of RNNs, in which the RNN can provide more precise forecasts
based on current data but cannot anticipate the word hold on in the long-term
memory. By default, LSTM will hold onto the information for a long time. On
the basis of time-series data, it is utilized for processing, predicting, and
categorizing [30].

The issue with stock market forecasting is that it relies heavily on
historical data, huge numbers, and a significant volume of data. As a result,
LSTM [31]helps RNNs by storing data for subsequent usage, which helps to
control error. As the stage goes on, the forecast gets more accurate proving
that it is far more reliable than other methods. To speed up training and
prevent overfitting,[32] suggested an LSTM model with two stacked LSTM layers,
an output value of 256, and a dropout ratio of 0.3. The stacked LSTM model
received a test score of 0.00875 MSE.

To enhance the prediction, [33] has taken use of the relationship between
stock price and sentiment. Use of live stock prices from the Yahoo Finance
API, web headlines, historical stock data, and sentiments from Twitter. The
univariate and multivariate LSTM were both fed sentiment scores from twitter,
the web, and Google Trend. Through the use of Weighted Average Ensemble, the
models were combined. As the days passed, the accuracy fell. The highest
prediction score accuracy was 0.99 for the first day, 0.92 for seven days, and
thereafter 0.62 until the thirty-first day.

  3. (c)

RNN: RNNs are a type of neural network that are constructed to handle
sequential input [26]. They are composed of regular recurrent cells, in which
the networks between the neurons form a directed graph, or, to put it another
way, the hidden layers contain a self-loop. This enables RNNs to use both the
information they have acquired in the past and the present input to learn the
current state by utilizing the hidden neurons’ previous state [31]. They can
pick up a variety of skills because of this. They include things like speech
and handwriting recognition. Chen, W. proposed an RNN-Boost model that made
use of the technical indicators and sentiment features, and Latent Dirichlet
allocation (LDA) features to predict the price of stocks. Its results showed
that the proposed model outperformed the single-RNN model [34]. RNN was used
[4] to create a model to forecast fuel costs in Chennai, Kolkata, Mumbai, and
Delhi for the next day. The past prices in these cities are kept up to date
using the international Brent oil barrel price and currency rates, and the
next day’s gasoline costs are forecasted. Based on their observations, they
concluded that the anticipated values are quite similar to the real prices.
For next-day pricing, the model’s accuracy is higher than 90

###  III-C Hybrid Models

Recent research has developed a method for mining vast volumes of data to draw
out useful information using a combination of several data mining techniques.
This section is a review of hybrid techniques that have been used to predict
petroleum prices. Hybrid models have a better prediction accuracy and are more
robust since they can combine the individual advantages of the techniques that
are used.

In Lake’s study, time-series data mining techniques were used to construct a
model for projecting retail oil and natural gas vehicle costs for vehicles in
Thailand. All three linear regressions, multi-layer perceptron, and support
vector machines for regression were employed [34]. The experiment’s time
series was separated into six data sets by the researcher. Gasohol 91, Gasohol
95, Gasohol E20, Gasohol E85, and Ultra Force Diesel and natural gas for
vehicles (NGV), with average prices of 30.7835-baht, 31.9928-baht,
28.6693-baht, 21.4488-baht, 27.4076-baht, and 12.1327-baht, respectively.

There were two parts to the research: a) For assessing the model’s efficiency,
use Root Mean Square Error (RMSE) and Mean Absolute Error (MAE). b) evaluating
the model’s performance using the Magnitude of Relative Error (MRE), which was
separated into monthly tests. The following were the outcomes: a) For gasoline
E85 and ultra-high-pressure diesel, linear regression showed the best fit,
with MMRE rates of 2.46 and 4.60, respectively. b) For gasohol 91, gasohol 95,
gasohol E20, and NGV, which had MMRE rates of 3.69, 3.20, 3.54, and 6.89,
respectively, the support vector machine for regression was the most
appropriate.

A hybrid model that combines data mining techniques such as nonlinear
autoregressive (NAR) neural network, autoregressive integrated moving average
model (ARIMA)and exponential smoothing model (ESM) was implemented in a study
done by [9]. These data mining techniques were combined in a state-space model
framework with the aim of increasing the accuracy of prediction of prices. The
three models, ESM, ARIM and NAR are combined in a linear form. The next step
is to determine the weights of the hybrid model using three approaches; use of
constant equal weights, use of constant genetic algorithm weights and use of
time varying weights[36]. The model is finally evaluated using mean absolute
error (MAE), root mean-square error (RMSE), mean absolute percentage error
(MAPE) and directional accuracy. From the study the hybrid model was able to
outperform its constructive models.

###  III-D Fuzzy Set and Logic

Instead of the traditional ”true or untrue” (1 or 0) Boolean operators on
which the contemporary computer is built, fuzzy logic bases computation on
”degrees of truth.” In the 1960s, Lotfi Zadeh of the University of California,
Barkley, was the first to put forth the concept of fuzzy logic [37]. On the
other hand, a class of items with a range of membership grades is referred to
as a fuzzy set. A membership (character) algorithm that awards each object a
membership grade ranging from zero to one defines such a set. Fuzzy logic and
fuzzy set are widely applied in counterparty society in the prediction of
commodities such as petroleum and gases by oil exporting companies and
manufacturers [8].

  1. (a)

OPEC Oil Price Prediction Using ANFIS: : In this study, the Organization of
Petroleum Exporting Countries (OPEC)’s oil prices are forecasted using an
adaptive neuro-fuzzy inference system (ANFIS) (OPEC). The suggested feature
set given to the ANFIS is the innovative part of the proposed model [37]. The
proposed approach is examined in the numerical models using an example of
simulating the time series of OPEC oil. In comparison to traditional neural
networks, ANFIS with the suggested variable set exhibits greater accuracy in
predicting oil prices.

  2. (b)

Forecasting the Direction of Short-Term Crude Oil Price Changes with Genetic-
Fuzzy Information Distribution: The short-term oil price forecasting challenge
is addressed in this research using a unique approach [38]. This method uses a
new fusion genetic-fuzzy information sharing method to perform the short-term
oil price series’ association with incomplete fuzzy information. Next, a
workable coding method based on multidimensional knowledge controlling points
is used to adapt the genetic-fuzzy dissemination of information to time series
forecasting. The empirical study findings show that the innovative fused
genetic-fuzzy information sharing approach statistically beats the standard
logistic regression model in terms of prediction accuracy using the market
prices of West Texas Medium (WTI) and London crude oil as sample data. The
outcomes show that this new strategy is successful in terms of direction
accuracy.

In a study carried out by [8], a model predicting crude oil price behavior in
the future was developed. The model included data from the previous year as
well as the actual oil price from the day before, allowing for more precise
forecasting. Their two-phase concept was built around fuzzy time series. In
the second step, they used the largest integer function to combine the
forecasts from their series and the actual crude oil prices from the previous
day. Surprisingly, the model captured the chaotic and non-linear behavior of
daily crude oil prices for 2021. ANN and SVM, on the other hand, are known to
provide superior outcomes in crude oil price prediction than traditional
approaches, thus these models were employed to statistically compare them to
their model. The suggested model outperformed existing models in terms of both
the RMSE and the MAE values, notably ANN and SVM.

TABLE II: comparison of the RMSE and MAE

##  IV The critiques of the existing models.

Numerous issues can be discovered in light of the models that has been
evaluated so far. First off, additional market-aligned inputs are not taken
into account because the data used in the projections is often derived from
the WTI or Brent price. Because of its reliance on a number of variables, the
market for petroleum prices is prone to volatility[38]. A prediction model
credibility might be damaged by ignoring these aspects, which prevents it from
being complete [33]. A model that produces accurate predictions shows strong
linkages between inputs and outputs, indicating the presence of dependency
[8].

Second, there are still few studies that focus on the market’s volatility. The
majority of discussions have centered more on the price side of the forecast
than on the causes of the fluctuations[7]. Although the supply and demand of
oil both significantly affect price volatility, the inclusion of these facts
merely restricts the potential of other elements, such as input data, making a
model incomplete [5]. An accurate market prediction may be made by integrating
and linking the important contributing components.

Third, time-series data have been used in the majority of the models examined
[9]. In the majority of the research, there was no data pretreatment or data
representation method. These two procedures aid in normalizing data sets,
removing noise from them, and cleaning up data sets to assist condense the
prediction process and ultimately produce reliable results[16]. The prediction
model will be less reliable without these operations.

Fourth, research has revealed that forecasting price trends is more common
than forecasting discrete prices[27]. Even if the applicability of the studies
completed so far is still debatable, discrete price forecasts will make
research more appealing and useful for practitioners.

##  V CONCLUSION AND FUTURE WORK

The vast amount of prior research has primarily employed quantitative
analytical techniques. This is due to the availability of historical market
data and recent developments in statistical techniques[27]. Most of these
studies have placed a significant reliance on the publication of official
macroeconomic statistics, which are gathered, examined, and aggregated by
regulatory agencies[8]. Politics, natural disasters, and emergent occurrences
have not previously been studied in the context of quantifiable evidence.
Although it comes with more complexity, including qualitative data into
petroleum predictions sounds promising. Compared to official statistics,
information on the internet is updated more regularly. For locating viewpoints
and obtaining information, text mining techniques are helpful. The development
of more sophisticated forecasting methods is necessary to increase the
accuracy of petroleum forecasting. These new data sources must be integrated
with high-frequency and lag-free data[27].

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