# Classification of the Lie and Noether symmetries for the Klein-Gordon
equation in Anisotropic Cosmology

Andronikos Paliathanasis
Institute of Systems Science, Durban University of Technology
PO Box 1334, Durban 4000, Republic of South Africa
Departamento de Matemáticas, Universidad Católica del Norte,
Avda. Angamos 0610, Casilla 1280 Antofagasta, Chile Email:
<EMAIL_ADDRESS>
###### Abstract

We carried out the detailed group classification of the potential in Klein-
Gordon equation in anisotropic Riemannian manifolds. Specifically, we consider
the Klein-Gordon equations for the four-dimensional anisotropic and
homogeneous spacetimes of Bianchi I, Bianchi III and Bianchi V. We derive all
the closed-form expressions for the potential function where the equation
admits Lie and Noether symmetries. We apply previous results which connect the
Lie symmetries of the differential equation with the collineations of the
Riemannian space which defines the Laplace operator, and we solve the
classification problem in a systematic way. Keywords: Lie symmetries; Klein-
Gordon; Anisotropic Spacetimes; Noether symmetries; Conformal Killing Vectors

##  1 Introduction

A systematic approach for the study of nonlinear differential equations is the
Lie symmetry analysis [1, 2, 3, 4]. The novelty of the Lie symmetry approach
is that through a systematic approach the existence of invariant
transformations can be determined. The latter can be used to simplify the
given differential equation with the use of similarity transformations. Under
the application of similarity transformations in a given differential
equation, we derive a new differential equation with less independent
variables. Furthermore, conservation laws can construct which are essential
for the study of the properties for the given differential equation [1]. Lie
symmetries have been applied for the study of nonlinear differential equations
in all areas of applied mathematics [5, 6, 7, 8, 9, 10, 11, 12, 13].

A systematic approach for the construction and the determination of
conservation laws for differential equations was established by E. Noether. In
Noether’s famous work of 1918 [14], Noether showed that some of the Lie
symmetries were related to symmetries of the variational principle. For each
symmetry of the variation integral, Noether derived an exact formula for the
derivation of the conservation law. That very simple method for the
construction of conservation laws is very important in physical science and in
other theories of applied mathematics.

In General Relativity the natural space is of four-dimensions described by a
Riemannian manifold. In this work we investigate the Lie and Noether
symmetries for the Klein-Gordon equation in anisotropic homogeneous
geometries. Anisotropic homogeneous spacetimes are of special interest because
they can describe the very early period of the universe, that is, before the
inflationary era where anisotropies played an important role in the evolution
of the physical variables. There is a plethora of studies in literature where
symmetry analysis has been applied for the classification of the geodesic
equations [15, 16, 17, 18], the wave equation [19, 20] in curved spaces and
the gravitational field equations [21, 22, 23, 24].

There has been investigated a relation between the symmetries of some
differential equations of special interest and the collineations of the
background geometry which provides the related differential operators. Indeed,
the Lie symmetries of the geodesic equations in Riemannian manifolds are
constructed by the elements of the projective group of the background
spacetime [25, 26]. The latter relation is true if we consider the existence
of a force term in the field equations [27]. For the Noether symmetries of the
geodesic Lagrangian these are derived by the elements of the homothetic group
of the background geometry [26]. That geometric results are extended and for
higher-order symmetries, see for instance the discussion in [28] and
references therein. As far as the case of partial differential equations is
concerned, the symmetries of the Poisson equation are constructed by the
elements of the conformal algebra of the Riemann metric which defines the
Laplace operator [29]. Hence, it is clear that in order to solve the
classification problem of our study we should present in detail the
classification of the conformal algebra for the homogeneous and anisotropic
spacetimes of our consideration. The structure of the paper is as follows.

In Section 2 we present in detail the theory of infinitesimal transformation
and the definitions of basic motions in Riemannian manifolds. Moreover, we
present the classification of the Killing symmetries, the Homothetic vector
and the proper Conformal Killing vector for the Bianchi I, Bianchi III and
Bianchi V spacetimes. In Section 3 we present the basic elements of the theory
of differential equations. For the Poisson and the Klein-Gordon equation we
recover previous results which show how the Lie symmetries are constructed
directly from the Conformal Killing vectors of the background geometry.
Moreover, a similar result is also presented and for the Noether symmetries of
the Klein-Gordon equation. The classification problem of our study is solved
in Section 4. We present all the functional forms of the potential function
for the Klein-Gordon equation where nontrivial symmetry vectors exist for the
Klein-Gordon equation. Finally, in Section 5 we summarise our results.

##  2 Infinitesimal Transformations and Motions of Riemannian spaces

Assume the Riemannian manifold $V^{n}$, $\dim V^{n}=n,$ with metric tensor
$g_{\mu\nu}$. Consider now the one-parameter point transformation defined by
the parametric equation
$\bar{x}^{\mu}=\bar{x}^{\mu}\left(x^{\nu},\varepsilon\right)~{}$which defines
a group orbit through the point $P\left(x^{\mu},0\right)$. Thus, the tangent
vector at the point $P$ is given by the following expression

$X=\frac{\partial\bar{x}^{\mu}}{\partial\varepsilon}|_{\varepsilon\rightarrow
0}\partial_{x^{\mu}}|_{P}.$ (1)

$X$ is the generator vector of the infinitesimal transformation near the point
$P$

$\bar{x}^{\mu}=x^{\mu}+\varepsilon\xi^{\mu}\left(x^{\nu}\right),$ (2)

in which
$\xi^{\mu}=\frac{\partial\bar{x}^{\mu}}{\partial\varepsilon}|_{\varepsilon\rightarrow
0}.$

Let $F\left(x^{\mu}\right)$ be a function in the Riemannian manifold defined
at the point $P$. Hence, under the action of the one-parameter point
transformation (1) the function reads $\bar{F}\left(\bar{x}^{\mu}\right)$.

By definition, function $F$ is invariant under the action of the one parameter
point transformation (1) if and only if it has the same value/expression
before and after the transformation. That is,
$\bar{F}\left(\bar{x}^{\mu}\right)=F\left(x^{\mu}\right)$ or equivalently
$\bar{F}\left(\bar{x}^{\mu}\right)=0$ when$~{}F\left(x^{\mu}\right)=0\,$. The
latter definition is described by the mathematical expression with the use of
the infinitessimal generator

$X\left(F\right)=0.$ (3)

equivalently $\xi^{\mu}\frac{\partial F}{\partial x^{\mu}}=0$.

Expression (3) is the Lie symmetry condition for a function
$F\left(x^{\mu}\right)$ to be invariant under the action of an one-parameter
point transformation in the base manifold. If condition (3) is true for a
specific vector field $X$, then $X$ is a Lie symmetry vector for the function
$F\left(x^{\mu}\right)$.

Consider now $\Omega^{\mu}\left(x^{\nu}\right)$ to be a geometric object with
the generic transformation rule [30]
$\bar{\Omega}^{\mu\prime}=\Phi^{\mu}\left(\Omega^{\nu},x^{\nu},\bar{x}^{\nu}\right).$When
$\Omega^{\mu}\left(x^{\nu}\right)$ is a linear homogeneous geometric object
the transformation rule reads [30]
$\Phi^{\mu}\left(\Omega^{\nu},x^{\nu},\bar{x}^{\nu}\right)=J_{\lambda}^{\mu}\left(x^{\nu},\bar{x}^{\nu}\right)\Omega^{\lambda}.$Where
$J_{\lambda}^{\mu}\left(x^{\nu},\bar{x}^{\nu}\right)$ is the Jacobian matrix
for the one-parameter point transformation with generator (1), that is
$J_{\lambda}^{\mu}=\frac{\partial\bar{x}^{\mu}}{\partial x^{\nu}}$.

Similar, with the definition of functions, the a geometric object
$\Omega^{\mu}\left(x^{\nu}\right)~{}$is invariant under a one parameter point
transformation (3) if and only if
$\bar{\Omega}^{\mu}\left(\bar{x}^{\nu}\right)=\Omega^{\mu}\left(x^{\nu}\right)$
or

$\mathcal{L}_{X}\Omega^{\mu}\left(x^{\nu}\right)=0,$ (4)

where $\mathcal{L}_{X}$ is the Lie derivative with respect to the vector field
$X$. In the case where $\Omega^{\mu}\left(x^{\nu}\right)\,$is a function for
the Lie derivative it holds $\mathcal{L}_{X}\Omega\equiv
X\left(\Omega\right)$.

A more generalized concept of the Lie symmetries for geometric objects are
summarized in the context of collineations. Consider now that for the
geometric object $\Omega$, the following expression holds

$\mathcal{L}_{X}\Omega=\Psi$ (5)

where $\Psi$ is a tensor field and it has the same components and symmetries
of the indices with $\Omega.~{}$If condition (5) is true ,$X$ is called a
collineation for the geometric object $\Omega,$ then, the type of
collineations is being defined by tensor field $\Psi$.

The metric tensor $g_{\mu\nu}$ of the Riemannian manifold is a linear
homogeneous geometric object with definition for the Lie derivative

$\mathcal{L}_{X}g_{\mu\nu}=X_{\left(\mu;\nu\right)}$ (6)

where ; denotes covariant derivative with respect to the Levi-Civita
connection.

For the metric tensor, the concept of collineations is expressed as

$\mathcal{L}_{X}g_{\mu\nu}=2\psi g_{\mu\nu}+2H_{\mu\nu}$ (7)

where $\psi$ is the conformal function and $H_{\mu\nu}$ is a symmetric
traceless tensor, i.e. $H_{~{}\mu}^{\mu}=0$. The most important collineations
for the metric tensor are the motions with $H_{\mu\nu}=0$. These are the
Killing vectors, the Homothetic vectors and the Conformal killing vectors.

The generator (1) of the infinitesimal transformation (2) is called a Killing
vector field (KV) for the Riemann space $V^{n},$ if and only if the metric
tensor is invariant under the action of the transformation, that is,

$\mathcal{L}_{X}g_{\mu\nu}=0.$ (8)

Moreover, the infinitesimal generator $X$ is a Conformal Killing vector (CKV)
for the Riemann space $V^{n}$ if there exists a function
$\psi\left(x^{\mu}\right)$ such that

$\mathcal{L}_{X}g_{\mu\nu}=2\psi g_{\mu\nu}$ (9)

where $\psi=\frac{1}{n}X_{;\mu}^{\mu}$.

An important class of collineations is when $\psi$ is a constant, then the CKV
becomes a Homothetic Killing Vector (HV). Moreover, when $\psi_{;\mu\nu}=0$,
the vector field $X$ is a special CKV (sp. CKV) for the Riemann manifold.
Indeed, when $\psi=0$, the CKV is also a KV. With the term proper CKV we shall
refer to CKVs which are not HVs or KVs.

The KVs, the HV and the CKVs form Lie algebras which are known as Killing
algebra $\left(G_{KV}\right)$, Homothetic algebra $\left(G_{HV}\right)~{}$and
Conformal Killing algebra $\left(G_{CV}\right)$. When for the dimensional of
the Riemannian manifold $V^{n}$ holds $n\geq 2,$ then $G_{KV}$ is a subalgebra
of $G_{HV}$ and the latter is a subalgebra of the Conformal Killing algebra,
that is $G_{KV}$ $\subseteq G_{HV}\subseteq G_{CV}$. For any Riemannian
manifold, there exists at most one proper Homothetic vector. Moreover, the
maximum dimensional Killing algebra is of $\frac{1}{2}n\left(n+1\right)$ and
the maximum Conformal Killing algebra is of
$\frac{1}{2}\left(n+1\right)\left(n+2\right)$ dimension.

Point transformations with a KV generator have the property to keep invariant
the length and the angles of autoparallels, unlike of the homothetic vector
where the angles remain invariant and the length is scaled with a constant
parameter. However, in the case where the point transformation is generated by
a CKV only the angles of autoparallels remain invariant.

The existence of collineations for the metric tensor is essential for the
nature of the physical space which is described by Riemannian geometry.
Indeed, our universe in large scales is described by the
Friedmann–Lemaître–Robertson–Walker line element which has a maximal symmetric
three-dimensional hypersurface. An important class of exact solutions in
General Relativity are the self-similar spacetimes. This family of solutions
has the main property to map to itself after an appropriate scale of the
dependent or independent variables, thus a proper HV exists. Self-similar
solutions of exact spacetimes describe the asymptotic behaviour of the most
general solution of the gravitational theory [31, 32]. Spacetimes with proper
CKV are also of special interest; more details can be found in [33].

###  2.1 CKVs of Anisotropic spacetimes

The Bianchi spacetimes describe anisotropic homogeneous cosmologies and they
are of special interest, because they can describe the very early stage of the
evolution of universe. In this family of spacetimes the line element of the
metric tensor is foliated along the time axis, with three dimensional
homogeneous hypersurfaces. The classification problem of all three dimensional
real Lie algebras was solved by Bianchi and it has shown that there are nine
Lie algebras. Thus, there are nine Bianchi models according to the admitted
Killing algebra of the three-dimensional homogeneous hypersurface.

The generic line element for the Bianchi model is

$ds^{2}=-N^{2}\left(t\right)dt^{2}+A^{2}(t)(\omega_{1})^{2}+B^{2}(t)(\omega_{2})^{2}+C^{2}(t)(\omega_{3})^{2}\,\text{.}$
(10)

where $\omega_{i}$, $i=1,2,3$, are basic one-forms and
$N\left(t\right),~{}A(t)$, $B(t)$, $C(t)$ are functions which depend only on
the time parameter, see [34]. In this study we are interested in the Bianchi
I, Bianchi III and Bianchi V spacetimes. These spacetimes in terms of the
coordinate expressions are diagonal.

Indeed, for these spacetimes the $1-$forms are

Bianchi I $\displaystyle:$
$\displaystyle\omega_{1}=dx~{},~{}\omega_{2}=dy~{},~{}\omega_{3}=dz~{},$
Bianchi III :
$\displaystyle\omega_{1}=dx~{},~{}\omega_{2}=dy~{},~{}\omega_{3}=e^{-x}dz~{},$
Bianchi V :
$\displaystyle\omega_{1}=dx~{},~{}\omega_{2}=e^{x}dy~{},~{}\omega_{3}=e^{x}dz~{}.$

The Killing algebras of the Bianchi spacetimes are presented in [34]. However,
the proper CKVs for the Bianchi I, Bianchi III and Bianchi V spacetimes have
been derived before in [35, 36].

The Bianchi I spacetime admits proper CKV when the metric tensor provides the
line element

$ds^{2}=C^{2}(t)\left[-dt^{2}+e^{-\frac{2}{c}t}dx^{2}+e^{-\frac{2\alpha_{1}}{c}t}dy^{2}+dz^{2}\right]$
(11)

with proper CKV the vector field

$X_{1}=c\partial_{t}+x\partial_{x}+\alpha_{1}y\partial_{y}$ (12)

and conformal factor $\psi(X_{1})=c\left(\ln\left|C\right|\right)_{,t}$. When
$C\left(t\right)=e^{\psi_{0}t}$, $X_{1}$ reduces to a proper HKV, while for
$C\left(t\right)=C_{0}$, $X_{1}$ is a KV.

Moreover, when the line element is of the form

$ds^{2}=C^{2}(t)\left[-dt^{2}+t^{2\frac{\alpha_{2}-1}{\alpha_{2}}}dx^{2}+t^{2\frac{\alpha_{2}-\alpha_{1}}{\alpha_{2}}}dy^{2}+dz^{2}\right]$
(13)

the resulting CKV is

$X_{2}=\alpha_{2}t\partial_{t}+x\partial_{x}+\alpha_{1}y\partial_{y}+\alpha_{2}z\partial_{z}$
(14)

with corresponding conformal factor
$\psi(X_{2})=\alpha_{2}\left[1+t(\ln\left|C\right|)_{,t}\right]$. Indeed, when
$C\left(t\right)=t^{\psi_{0}-1}$, $X_{2}$ is a proper HKV, while when
$C\left(t\right)=t^{-1}$, $X_{2}$ is reduced to a KV.

The Bianchi III spacetime admits a proper CKV when

$ds^{2}=A^{2}\left(t\right)\left[e^{m\lambda
t}\left(-dt^{2}+dx^{2}\right)+e^{m\left(\lambda-1\right)t}dy^{2}+e^{-2x}dz^{2}\right].$
(15)

where now the corresponding vector field is

$X_{3}=\frac{2}{m}\partial_{t}+y\partial_{y}+\lambda z\partial_{z}$ (16)

and $\psi_{(III)}(X_{3})=\frac{2}{m}\frac{A_{,t}}{A}+\lambda$. Indeed, when
$A\left(t\right)=e^{A_{0}t}$, the vector field is reduced to a HV, and for
$A\left(t\right)=e^{-\frac{\lambda}{2}mx}$, is reduced to a KV.

Finally, for the family of Bianchi V spacetimes it follows that the line
element

$ds^{2}=A^{2}\left(t\right)\left[e^{m\lambda
t}\left(-dt^{2}+dx^{2}\right)+e^{2x}\left(e^{m(\lambda-1)t}dy^{2}+dz^{2}\right)\right]$
(17)

admits as proper CKV the vector field $X_{3}$ with the same conformal factor
as before.

Recall that in the following we shall not investigate the case where the
spacetimes reduce to locally rotational spaces or the scale factors are
constant functions.

##  3 Symmetries of differential equations

In terms of geometry a differential equation can be considered as a function
$H=H(x^{\nu},u^{A},u_{,\mu}^{A},u_{,\mu\nu}^{A})$ in the space
$B=B\left(x^{\nu},u^{A},u_{,\mu}^{A},u_{,\mu\nu}^{A}\right)$,
$u^{A}=u^{A}\left(x^{\mu}\right)$ denote the dependent variables, $x^{\mu}$
are the independent variables and $u_{,\mu}^{A}=\frac{\partial u^{A}}{\partial
u^{\mu}}$.

Assume now the infinitesimal transformation in the base manifold of the
differential equation $H$,

$\displaystyle\bar{x}^{\mu}$
$\displaystyle=x^{\mu}+\varepsilon\xi^{\mu}(x^{\nu},u^{B})~{},$ (18)
$\displaystyle\bar{u}^{A}$
$\displaystyle=\bar{u}^{A}+\varepsilon\eta^{A}(x^{\nu},u^{B})~{},$ (19)

with vector generator

$\mathbf{X}=\xi^{\mu}(x^{\nu},u^{B})\partial_{x^{\mu}}+\eta^{A}(x^{\nu},u^{B})\partial_{u^{A}}~{}.$
(20)

Similarly to the case of functions, the geometric vector field $X\mathbf{\
}$is a Lie symmetry of $H=H(x^{\nu},u^{A},u_{,\mu}^{A},u_{,\mu\nu}^{A})$ if
and only if the following is true [3, 2]

$X^{[2]}(H)=0,$ (21)

in which $X^{[2]}$ is the second extension of the vector field $X$ in the
space$~{}B=B\left(x^{\nu},u^{A},u_{,\mu}^{A},u_{,\mu\nu}^{A}\right)$ defined
as

$X^{[2]}=X+\eta_{\mu}^{A}\partial_{u_{\mu}^{A}}+\eta_{\mu\nu}^{A}\partial_{u_{\mu\nu}^{A}}$
(22)

in which

$\eta_{\mu}^{A}=D_{\mu}\eta^{A}-D_{\mu}\xi^{\nu}u_{,\nu}^{A}~{},$ (23)

and

$\eta_{\mu\nu}^{A}=D_{\nu}\eta_{\mu}^{A}-D_{\nu}D_{\mu}\xi^{\kappa}u_{,\kappa}^{A}~{},$

where $D_{\mu}~{}$is the total derivative.

A straightforward application of the Lie symmetries for a given differential
equation is the construction of invariant functions by deriving the
characteristic functions. The characteristic functions can be used to define
similarity transformations which can be used to reduce the number of the
indepedent variables in the case of partial differential equations.

The invariants are determined by the solution of the following Lagrangian
system.

$\frac{dx^{\mu}}{\xi^{\mu}}=\frac{du^{A}}{\eta^{A}}=\frac{du_{\mu}^{A}}{\eta_{\mu}^{A}}=\frac{du_{\mu\nu}^{A}}{\eta_{\mu\nu}^{A}}.$
(24)

In the case where the differential equation $H$ follows from a variational
principle with Lagrangian function $L=L(x^{\mu},u^{A},u_{,\mu}^{A})$ such as
$H\equiv\mathbf{E}\left(L\right)=0$, where $\mathbf{E}$ is the Euler operator.
The Lie point symmetry $\mathbf{X}$ of the DE $H\ \ $is a Noether point
symmetry of $H,\ $if and only if the following condition is satisfied

$\mathbf{X}^{[1]}L+LD_{i}\xi^{i}=D_{i}A^{i}\left(x^{k},u^{C}\right),~{}$ (25)

where $\mathbf{X}^{[1]}$ is the first prolongation of $\mathbf{X,}$ and
$A^{\mu}$ is a vector field which should be determined. Condition (25) is
Noether’s second theorem. The second theorem of Noether states that for any
vector field $X$ where condition (25) is true the following function is a
conservation law

$I^{\mu}=\xi^{\nu}\left(u_{\nu}^{A}\frac{\partial L}{\partial
u_{\mu}^{A}}-\delta_{\nu}^{\mu}L\right)-\eta^{A}\frac{\partial L}{\partial
u_{\mu}^{A}}+A^{\mu}~{},$ (26)

that is, $D_{\mu}I^{\mu}=0,~{}$[3, 2].

###  3.1 Poisson equation

Let $\Delta$ be the Laplace operator in the Riemannian manifold $V^{n}$,

$\Delta=\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x^{\mu}}\left(\sqrt{g}g^{\mu\nu}\frac{\partial}{\partial x^{\nu}}\right),$

then the Poisson equation reads

$\Delta u=f\left(x^{\mu},u\right),$ (27)

or equivalently

$g^{\mu\nu}u_{\mu\nu}-\Gamma^{\mu}u_{\mu}=f\left(x^{\nu},u\right),$ (28)

where $\Gamma^{\mu}=\Gamma_{\nu\kappa}^{\mu}g^{\nu\kappa}$ and
$\Gamma_{\nu\kappa}^{\mu}$ are the Christoffel symbols for the Levi-Civita
connection of the metric tensor $g_{\mu\nu}$.

The Lie symmetry analysis for the Poisson equation when $f=f\left(u\right)$
have been given in [37], and for $f=f\left(x^{i},u\right)$ are presented in
[29]. Indeed, the Lie (point) symmetries for the Poisson equation are related
to the elements of the conformal algebra for the Riemannian manifold as
described in the following.

###### Theorem 1

The Lie symmetries for the Poisson equation are constructed by the generic CKV
of the background metric tensor $g_{\mu\nu}$ of the Riemannian manifold
$V^{n}$:

a) For $n>2$, the Lie symmetry vector has the generic form

$X=\xi^{\mu}\left(x^{\nu}\right)\partial_{\mu}+\left(\frac{2-n}{2}\psi\left(x^{\nu}\right)u+a_{0}u+b\left(x^{\nu}\right)\right)\partial_{u},$
(29)

where $\xi^{i}\left(x^{\nu}\right)$ is a CKV of the Riemannian manifold with
conformal factor $\psi\left(x^{\nu}\right)$ and the following condition holds

$\frac{2-n}{2}\Delta\psi
u+g^{\mu\nu}b_{\mu;\nu}-\xi^{\nu}f_{,\nu}-\frac{2-n}{2}\psi
uf_{,u}-\frac{n+2}{2}\psi f-bf_{,u}=0,$ (30)

b) For $n=2$, the generic Lie symmetry vector is

$X=\xi^{\mu}\left(x^{\nu}\right)\partial_{\mu}+\left(a_{0}u+b\left(x^{\nu}\right)\right)\partial_{u},$
(31)

where $\xi^{\mu}$ is a CKV and the following conditions are satisfied

$g^{\mu\nu}b_{;\mu\nu}-\xi^{\nu}f_{,\nu}-a_{0}uf_{,u}+\left(a_{0}-2\psi\right)f-bf_{,u}=0,$
(32)

that is, the function $b$ is solution of the Laplace equation.

A special case of the Poisson equation is the Klein-Gordon equation with
$f\left(x^{\nu},u\right)=V\left(x^{\nu}\right)u$, that is,

$\Delta u-V\left(x^{\nu}\right)u=0,$ (33)

where $V(x^{\mu})$ is the potential function. For the Lie symmetries of the
Klein-Gordon equation it follows

###### Theorem 2

For the Klein Gordon equation (33) the Lie symmetries are constructed by the
elements of the conformal algebra of the Riemannian manifold:

a) for $n>2$, the generic symmetry vector is expressed as

$X=\xi^{\mu}\left(x^{\nu}\right)\partial_{\mu}+\left(\frac{2-n}{2}\psi\left(x^{\nu}\right)u+a_{0}u+b\left(x^{\nu}\right)\right)\partial_{u},$
(34)

where now $\xi^{\mu}$ is a CKV with conformal factor
$\psi\left(x^{\nu}\right),$ $b\left(x^{\nu}\right)$ solves equation (33) with
constraint condition

$\xi^{\nu}V_{,\nu}+2\psi V-\frac{2-n}{2}\Delta\psi=0,$ (35)

b) for $n=2$, the generic symmetry vector is written

$X=\xi^{\mu}\left(x^{\nu}\right)\partial_{\mu}+\left(a_{0}u+b\left(x^{\nu}\right)\right)\partial_{u},$
(36)

where $\xi^{\mu}$ is a CKV with conformal factor $\psi\left(x^{\nu}\right),$
$b\left(x^{\nu}\right)$ solves equation (33) with constrain

$\xi^{\nu}V_{,\nu}+2\psi V=0.$ (37)

The Klein-Gordon equation (33) can be reproduced by the variation of the
Lagrangian function

$L\left(x^{\nu},u,u_{,\nu}\right)=\frac{1}{2}\sqrt{\left|g\right|}g^{\mu\nu}u_{,\mu}u_{,\nu}+\frac{1}{2}\sqrt{\left|g\right|}V\left(x^{\nu}\right)u^{2}.$
(38)

Therefore, for the Noether symmetries of the Klein-Gordon Lagrangian (38) the
following Theorem holds.

###### Theorem 3

The Lie point symmetries of the Klein Gordon equation (33) are generated from
the elements of the conformal algebra of the Riemannian manifold, where the
generic Noether symmetry is of the form

$X_{N}=\xi^{\mu}\left(x^{\nu}\right)\partial_{\mu}+\left(\frac{2-n}{2}\psi\left(x^{\nu}\right)u\right)\partial_{u},$

where the corresponding vector
$A_{\mu}=\frac{2-n}{4}\sqrt{\left|g\right|}\psi_{,\mu}\left(x^{\nu}\right)u^{2}$,
in which $\xi^{\mu}\left(x^{\nu}\right)$ is a CKV with conformal factor
$\psi\left(x^{\nu}\right).$ The constraint equation is

$\xi^{\nu}V_{,\nu}+2\psi V-\frac{2-n}{2}\Delta\psi=0.$ (39)

We remark that for the Klein-Gordon equation all the non-trivial Lie
symmetries are also Noether symmetries. The resulting conservation law is of
the form

$I^{\mu}=\sqrt{\left|g\right|}\left(\left(\frac{1}{2}g^{\kappa\nu}u_{,\kappa}u_{,\nu}-\frac{1}{2}V\left(x^{\nu}\right)u^{2}\right)\xi^{\mu}-\eta\frac{1}{2}g^{\mu\nu}u_{,\nu}+\frac{2-n}{4}\psi_{,\mu}\left(x^{\nu}\right)u^{2}\right).$
(40)

##  4 Klein-Gordon equation in anisotropic geometries

We proceed with the solution of the classification problem for the potential
function $V\left(x^{\mu}\right)$ for the Klein-Gordon equation (33) in the
case of anisotropic cosmologies where the Klein-Gordon equation admits non-
trivial Lie symmetries. The trivial symmetries are the vector fields
$X_{u}=u\partial_{u}$, $X_{b}=b\partial_{u}$ which exist for any potential
function $V\left(x^{\nu}\right)$.

###  4.1 Bianchi I

In a Bianchi I spacetime, the Klein-Gordon equation is written

$\left(-\frac{u_{,tt}}{N^{2}}+\frac{u_{,xx}}{A^{2}}+\frac{u_{,yy}}{B^{2}}+\frac{u_{,zz}}{C^{2}}\right)+\frac{1}{N^{2}}\left(\frac{N_{.t}}{N}u_{,t}-\frac{A_{,t}}{A}u_{,x}-\frac{C_{,t}}{C}u_{,y}-\frac{C_{,t}}{C}u_{,z}\right)-\frac{V\left(t,x,y,z\right)}{A^{2}B^{2}C}u^{2}=0.$
(41)

For arbitrary function forms of the scale factors the background space admits
the three KVs, $\xi_{I}^{1}=\partial_{x}~{},~{}\xi_{I}^{2}=\partial_{y}$ and
$\xi_{I}^{3}=\partial_{z}$.

Hence, from Theorem 2 it follows that: (i) $\xi_{I}^{1}$, is a Lie symmetry
when $V_{i}^{I}\left(t,x,y,z\right)=V\left(t,y,z\right)$; (ii) $\xi_{I}^{2}$
is a Lie symmetry when $V\left(t,x,y,z\right)=V_{ii}^{I}\left(t,x,z\right)$;
(iii) $\xi_{I}^{3}$ is a Lie symmetry when
$V\left(t,x,y,z\right)=V_{iii}^{I}\left(t,x,y\right)$; and (iv)
$\alpha\xi_{I}^{1}+\beta\xi_{I}^{2}+\gamma\xi_{I}^{3}$ is a Lie symmetry when
$V_{iv}^{I}\left(t,x,y,z\right)=V\left(t,y-\frac{\beta}{\alpha}x,z-\frac{\gamma}{\alpha}x\right)$.

In the special case where the line element is that of (11) the CKV $X_{1}$
produces the Lie symmetry vector
$X=X_{1}+\left(-2c\left(\ln\left|C\right|\right)_{,t}u\right)\partial_{u}$ for
the Klein-Gordon equation, if and only if

$V_{v}^{I}\left(t,x,y,z\right)=\frac{cC_{,tt}-\left(\alpha_{1}+1\right)C_{,t}}{cC^{3}}+\frac{1}{C^{2}}U\left(xe^{-\frac{t}{c}},ye^{-\frac{a_{1}}{c}t},z\right).$
(42)

Similarly, the vector field
$X=X_{1}+\alpha\xi_{I}^{1}+\beta\xi_{I}^{2}+\gamma\xi_{I}^{3}+\left(-2c\left(\ln\left|C\right|\right)_{,t}u\right)\partial_{u}$
is a Lie symmetry for the Klein-Gordon equation in a Bianchi I spacetime with
line element (11) if and only if

$V_{vi}^{I}\left(t,x,y,z\right)=\frac{cC_{,tt}-\left(\alpha_{1}+1\right)C_{,t}}{cC^{3}}+\frac{1}{C^{2}}U\left(\left(x+\alpha\right)e^{-\frac{t}{c}},\left(y+\frac{\beta}{a_{1}}\right)e^{-\frac{a_{1}}{c}t},z-\frac{\gamma}{c}\tau\right).$
(43)

On the other hand, for the Bianchi I line element (13) the CKV $X_{2}$ is the
generator of the Lie symmetry vector
$X=X_{2}-\alpha_{2}\left[1+t(\ln\left|C\right|)_{,t}\right]u\partial_{u}$ for
the Klein-Gordon equation (41) for the potential function

$V_{vii}^{I}\left(t,x,y,z\right)=\frac{1}{t^{2}C^{2}}\left(U\left(xt^{-\frac{1}{a_{2}}},yt^{-\frac{a_{1}}{a_{2}}},\frac{z}{t}\right)+F\left(t\right)\right)$
(44)

where

$\displaystyle F\left(t\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{a_{2}C^{2}}\left(a_{2}t^{2}CC_{,ttt}+tC_{,tt}\left(a_{2}C_{,t}+C\left(a_{1}-4a_{2}+1\right)\right)\right)+$
(45)
$\displaystyle-\frac{1}{a_{2}C^{2}}\left(C_{,t}\left(1+a_{1}-2a_{2}\right)\left(C-tC_{,t}\right)\right)\text{\thinspace}.$

Moreover, the vector field
$X=X_{2}+\alpha\xi_{I}^{1}+\beta\xi_{I}^{2}+\gamma\xi_{I}^{3}-\alpha_{2}\left[1+t(\ln\left|C\right|)_{,t}\right]\partial_{u}$
is a Lie symmetry for the Klein-Gordon equation when

$V_{viii}^{I}\left(t,x,y,z\right)=\frac{1}{t^{2}C^{2}}\left(U\left(\left(x+\alpha\right)t^{-\frac{1}{a_{2}}},\left(y+\frac{\beta}{a_{1}}\right)t^{-\frac{a_{1}}{a_{2}}},\left(z+\frac{\gamma}{a_{2}}\right)t^{-1}\right)+F\left(t\right)\right).$
(46)

####  4.1.1 Invariant functions

Let us now determine the invariant functions which correspond to each admitted
Lie point symmetry. The invariant functions can be used to determine
similarity transformations whenever they are applied the number of dependent
variables of the Klein-Gordon equation is reduced.

For the vector field $\xi_{I}^{1}$, the invariant functions are
$\left\\{t,y,z,u\right\\}$. Similarly, for the vector field $\xi_{I}^{2}$ we
determine the Lie invariants $\left\\{t,x,z,u\right\\}$. Moreover, for
$\xi_{I}^{3}$ the Lie invariants are $\left\\{t,x,y,u\right\\}$, while for the
vector field $\alpha\xi_{I}^{1}+\beta\xi_{I}^{2}+\gamma\xi_{I}^{3}$, the Lie
invariants are
$\left\\{t,y-\frac{\beta}{\alpha}x,z-\frac{\gamma}{\alpha}x,u\right\\}$.

Furthermore, for the potential function $V_{v}^{I}$ where
$X=X_{1}+\left(-2c\left(\ln\left|C\right|\right)_{,t}u\right)\partial_{u}$ is
a Lie symmetry, the resulting Lie invariants are calculated
$\left\\{xe^{-\frac{t}{c}},ye^{-\frac{a_{1}}{c}t},z,uC\left(t\right)^{2}\right\\}$.
For the potential $V_{vi}^{I}$, the Lie invariants are
$\left\\{\left(x+\alpha\right)e^{-\frac{t}{c}},\left(y+\frac{\beta}{a_{1}}\right)e^{-\frac{a_{1}}{c}t},z-\frac{\gamma}{c}\tau,uC\left(t\right)^{2}\right\\}$.
In a similar way, for the potential functions
$V_{vii}^{I}\left(t,x,y,z\right)$ and $V_{viii}^{I}\left(t,x,y,z\right)$ the
admitted Lie invariants are
$\left\\{xt^{-\frac{1}{a_{2}}},yt^{-\frac{a_{1}}{a_{2}}},\frac{z}{t},ut^{2}C\left(t\right)^{2}\right\\}$and
$\left\\{\left(x+\alpha\right)t^{-\frac{1}{a_{2}}},\left(y+\frac{\beta}{a_{1}}\right)t^{-\frac{a_{1}}{a_{2}}},\left(z+\frac{\gamma}{a_{2}}\right)t^{-1},ut^{2}C\left(t\right)^{2}\right\\}$
provided by the Lie symmetries
$X_{2}-\alpha_{2}\left[1+t(\ln\left|C\right|)_{,t}\right]u\partial_{u}$ and
$X_{2}+\alpha\xi_{I}^{1}+\beta\xi_{I}^{2}+\gamma\xi_{I}^{3}-\alpha_{2}\left[1+t(\ln\left|C\right|)_{,t}\right]\partial_{u}$
respectively.

####  4.1.2 Conservation laws

We apply Noether’s theorem and expression (40) hence the resulting
conservation laws related to the admitted Lie symmetries for the Klein-Gordon
equation (41) are

$I^{x}\left(\xi_{I}^{1}\right)=\frac{NABC}{2}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{1}{B^{2}}u_{,y}^{2}+\frac{1}{C^{2}}u_{,z}^{2}\right)-V_{i}^{I}\left(t,x,y,z\right)u^{2}\right)\right).$
(47)
$I^{t}\left(\xi_{I}^{1}\right)=0~{},~{}I^{y}\left(\xi_{I}^{1}\right)=0\text{
and }I^{z}\left(\xi_{I}^{1}\right)=0.$ (48)

For the vector field $\xi_{I}^{2}$ there exists the conservation law

$I^{y}\left(\xi_{I}^{2}\right)=\frac{NABC}{2}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{1}{B^{2}}u_{,y}^{2}+\frac{1}{C^{2}}u_{,z}^{2}\right)-V_{ii}^{I}\left(t,x,y,z\right)u^{2}\right)\right).$
(49)
$I^{t}\left(\xi_{I}^{2}\right)=0~{},~{}I^{x}\left(\xi_{I}^{2}\right)=0\text{
and }I^{z}\left(\xi_{I}^{2}\right)=0.$ (50)

While for the vector field $\xi_{I}^{3}$ the resulting Noetherian conservation
law is

$I^{z}\left(\xi_{I}^{3}\right)=\frac{NABC}{2}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{1}{B^{2}}u_{,y}^{2}+\frac{1}{C^{2}}u_{,z}^{2}\right)-V_{iii}^{I}\left(t,x,y,z\right)u^{2}\right)\right).$
(51)
$I^{t}\left(\xi_{I}^{3}\right)=0~{},~{}I^{x}\left(\xi_{I}^{3}\right)=0\text{
and }I^{y}\left(\xi_{I}^{3}\right)=0.$ (52)

For the generic vector field
$\alpha\xi_{I}^{1}+\beta\xi_{I}^{2}+\gamma\xi_{I}^{3}$ we calculate the
conservation law

$I^{x}\left(\alpha\xi_{I}^{1}+\beta\xi_{I}^{2}+\gamma\xi_{I}^{3}\right)=\alpha\frac{NABC}{2}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{1}{B^{2}}u_{,y}^{2}+\frac{1}{C^{2}}u_{,z}^{2}\right)-V_{iv}^{I}\left(t,x,y,z\right)u^{2}\right)\right),$
(53)
$I^{y}\left(\alpha\xi_{I}^{1}+\beta\xi_{I}^{2}+\gamma\xi_{I}^{3}\right)=\beta\frac{NABC}{2}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{1}{B^{2}}u_{,y}^{2}+\frac{1}{C^{2}}u_{,z}^{2}\right)-V_{iv}^{I}\left(t,x,y,z\right)u^{2}\right)\right),$
(54)
$I^{z}\left(\alpha\xi_{I}^{1}+\beta\xi_{I}^{2}+\gamma\xi_{I}^{3}\right)=\gamma\frac{NABC}{2}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{1}{B^{2}}u_{,y}^{2}+\frac{1}{C^{2}}u_{,z}^{2}\right)-V_{iv}^{I}\left(t,x,y,z\right)u^{2}\right)\right)$
(55)
$I^{t}\left(\alpha\xi_{I}^{1}+\beta\xi_{I}^{2}+\gamma\xi_{I}^{3}\right)=0~{}.$
(56)

For the potential $V_{v}^{I}\left(t,x,y,z\right)$ there exists the
conservation law

$I^{t}\left(X_{1}\right)=c\frac{e^{-\frac{t\left(1+\alpha_{1}\right)}{c}}C^{4}}{2}\left(H_{v}^{I}-\left(\left(\ln\left|C\right|\right)_{,t}u\right)\frac{1}{C}u^{,t}-\frac{1}{2C}\left(\ln\left|C\right|\right)_{,tt}u^{2}\right).$
(57)
$I^{x}\left(X_{1}\right)=\frac{e^{-\frac{t\left(1+\alpha_{1}\right)}{c}}C^{4}}{2}\left(xH_{v}^{I}-\left(c\left(\ln\left|C\right|\right)_{,t}u\right)\frac{1}{C}u^{,x}\right).$
(58)
$I^{y}\left(X_{1}\right)=\frac{e^{-\frac{t\left(1+\alpha_{1}\right)}{c}}C^{4}}{2}\left(\alpha_{1}yH_{v}^{I}-\left(c\left(\ln\left|C\right|\right)_{,t}u\right)\frac{1}{C}u^{,y}\right).$
(59)
$I^{z}\left(X_{1}\right)=\frac{e^{-\frac{t\left(1+\alpha_{1}\right)}{c}}C^{4}}{2}\left(-\left(c\left(\ln\left|C\right|\right)_{,t}u\right)\frac{1}{N}u^{,z}\right).$
(60)

where

$H_{v}^{I}=\left(\left(-\frac{1}{C^{2}}u_{,t}^{2}+\frac{e^{\frac{2t}{c}}}{C^{2}}u_{,x}^{2}+\frac{e^{\frac{2\alpha_{1}}{c}t}}{C^{2}}u_{,y}^{2}+\frac{1}{C^{2}}u_{,z}^{2}\right)-V_{v}^{I}\left(t,x,y,z\right)u^{2}\right)$
(61)

For the potential function $V_{vi}^{I}\left(t,x,y,z\right)$ the resulting
conservation law is derived

$I^{t}=I^{t}\left(X_{1}\right),$ (62) $I^{x}=I^{x}\left(X_{1}\right)+\alpha
I^{x}\left(\xi_{I}^{1}\right)~{},$ (63) $I^{y}=I^{y}\left(X_{1}\right)+\beta
I^{y}\left(\xi_{I}^{1}\right)~{},$ (64) $I^{z}=I^{z}\left(X_{1}\right)+\gamma
I^{z}\left(\xi_{I}^{1}\right)~{},$ (65)

for
$N\left(t\right)=C\left(t\right),~{}A\left(t\right)=e^{-\frac{t}{c}}C\left(t\right)~{}$and
$B\left(t\right)=e^{-\frac{\alpha_{1}}{c}t}C\left(t\right)$ with potential
function $V_{vi}^{I}\left(t,x,y,z\right)$

For the potential function$V_{vii}^{I}$ where $X_{2}$ is the generator of the
Lie symmetry vector the resulting Noetherian conservation law is

$I^{t}\left(X_{2}\right)=\frac{C^{4}}{2}t^{2-\frac{1+\alpha_{1}}{\alpha_{2}}}\left(\alpha_{2}tH_{vii}^{I}-\alpha_{2}\left(\left(\ln\left|C\right|\right)_{,t}u\right)\frac{1}{C}u^{,t}-\frac{1}{2C}\alpha_{2}\left[1+t(\ln\left|C\right|)_{,t}\right]_{,t}u^{2}\right).$
(66)
$I^{x}\left(X_{2}\right)=\frac{C^{4}}{2}t^{2-\frac{1+\alpha_{1}}{\alpha_{2}}}\left(xH_{vii}^{I}-\left(\alpha_{2}\left[1+t(\ln\left|C\right|)_{,t}\right]u\right)\frac{1}{C}u^{,x}\right).$
(67)
$I^{y}\left(X_{2}\right)=\frac{C^{4}}{2}t^{2-\frac{1+\alpha_{1}}{\alpha_{2}}}\left(\alpha_{1}yH_{vii}^{I}-\left(\alpha_{2}\left[1+t(\ln\left|C\right|)_{,t}\right]u\right)\frac{1}{C}u^{,y}\right).$
(68)
$I^{z}\left(X_{2}\right)=\frac{C^{4}}{2}t^{2-\frac{1+\alpha_{1}}{\alpha_{2}}}\left(\alpha_{2}zH_{vii}^{I}-\left(\alpha_{2}\left[1+t(\ln\left|C\right|)_{,t}\right]u\right)\frac{1}{N}u^{,z}\right).$
(69)

where

$H_{vii}^{I}=\left(\left(-\frac{1}{C^{2}}u_{,t}^{2}+\frac{1}{C^{2}t^{2-\frac{2}{\alpha_{2}}}}u_{,x}^{2}+\frac{1}{C^{2}t^{2-\frac{2\alpha_{1}}{\alpha_{2}}}}u_{,y}^{2}+\frac{1}{C^{2}}u_{,z}^{2}\right)-V_{vii}^{I}\left(t,x,y,z\right)u^{2}\right).$
(70)

Finally, for the potential function $V_{viii}^{I}\left(t,x,y,z\right)$ the
conservation law is

$I^{t}=I^{t}\left(X_{2}\right),$ (71) $I^{x}=I^{x}\left(X_{2}\right)+\alpha
I^{x}\left(\xi_{I}^{1}\right)~{},$ (72) $I^{y}=I^{y}\left(X_{2}\right)+\beta
I^{y}\left(\xi_{I}^{1}\right)~{},$ (73) $I^{z}=I^{z}\left(X_{2}\right)+\gamma
I^{z}\left(\xi_{I}^{1}\right)~{},$ (74)

for
$N\left(t\right)=C\left(t\right),~{}A\left(t\right)=t^{1-\frac{1}{\alpha_{2}}}C\left(t\right)$
,$~{}B\left(t\right)=t^{1-\frac{\alpha_{1}}{\alpha_{2}}}C\left(t\right)$ and
potential function $V_{viii}^{I}\left(t,x,y,z\right)$

###  4.2 Bianchi III

In the Bianchi III geometry, the Klein-Gordon equation reads

$\left(-\frac{u_{,tt}}{N^{2}}+\frac{u_{,xx}}{A^{2}}+\frac{u_{,yy}}{B^{2}}+e^{2x}\frac{u_{,zz}}{C^{2}}\right)+\frac{1}{N^{2}}\left(\frac{N_{.t}}{N}u_{,t}-\frac{A_{,t}}{A}u_{,x}-\frac{C_{,t}}{C}u_{,y}-\frac{C_{,t}}{C}u_{,z}\right)-\frac{1}{A^{4}}u_{,x}-\frac{V\left(t,x,y,z\right)}{A^{2}B^{2}C}u^{2}=0.$
(75)

The three KVs of the Bianchi III spacetime are
$\xi_{III}^{1}=\partial_{x}+z\partial_{z},~{}\xi_{III}^{2}=\partial_{y}$ and
$\xi_{III}^{3}=\partial_{z}$. Hence, (i) $\xi_{III}^{1}$ is a Lie symmetry for
equation (75) when
$V_{i}^{III}\left(t,x,y,z\right)=V\left(t,y,ze^{-x}\right)$; (ii)
$\xi_{III}^{2}$ is a Lie symmetry for
$V_{ii}^{III}\left(t,x,y,z\right)=V\left(t,x,z\right)$; (iii) $\xi_{III}^{3}$
is a Lie symmetry when
$V_{iii}^{III}\left(t,x,y,z\right)=V\left(t,x,y\right)$; (iv)
$\alpha\xi_{III}^{1}+\beta\xi_{III}^{2}+\gamma\xi_{III}^{3}$ is a Lie symmetry
when
$V_{iv}^{III}\left(t,x,y,z\right)=V\left(t,y-\frac{\beta}{\alpha}x,\left(z+\frac{\gamma}{\alpha}\right)e^{-x}\right)$.

For the line (15) where the Bianchi III spacetime admits the additional CKV
$X_{3}$, it follows that the vector field
$X=X_{3}-\left(\frac{2}{m}\frac{A_{,t}}{A}+\lambda\right)u\partial_{u}$ is a
Lie symmetry vector for the Klein-Gordon equation (75) when

$V_{v}^{III}\left(t,x,y,z\right)=\frac{1}{A^{2}}e^{-m\lambda
t}U\left(x,ye^{-\frac{m}{2}t},ze^{-\frac{m\lambda}{2}t}\right)+\frac{m\left(\lambda-1\right)A_{,t}+2A_{,tt}}{2A^{3}}\text{.}$
(76)

Hence, the vector field
$X=X_{3}+\alpha\xi_{III}^{1}+\beta\xi_{III}^{2}+\gamma\xi_{III}^{3}-\left(\frac{2}{m}\frac{A_{,t}}{A}+\lambda\right)u\partial_{u}$
is a Lie symmetry of equation (75) for the potential function

$V_{vi}^{III}\left(t,x,y,z\right)=\frac{1}{A^{2}}e^{-m\lambda
t}U\left(x-\frac{\alpha
m}{2}t,\left(y+\beta\right)e^{-\frac{m}{2}t},\left(z+\frac{\gamma}{\lambda+\alpha}\right)e^{-\frac{m\left(\lambda+\alpha\right)}{2}t}\right)+\frac{m\left(\lambda-1\right)A_{,t}+2A_{,tt}}{2A^{3}}.$
(77)

####  4.2.1 Invariant functions

We proceed with the derivation of the invariant functions provided by each
case for the above potential functions. For $V_{i}^{III}\left(t,x,y,z\right)$
the Lie invariants are $\left\\{t,y,ze^{-x},u\right\\}$, for
$V_{ii}^{III}\left(t,x,y,z\right)$ and the Lie symmetry vector $\xi_{III}^{2}$
we determine the Lie invariants $\left\\{t,x,z,u\right\\}$ while from
$\xi_{III}^{3}$ for the potential $V_{iii}^{III}\left(t,x,y,z\right)$ the Lie
invariants are $\left\\{t,x,y,u\right\\}$. Similarly, for the generic vector
field $\alpha\xi_{III}^{1}+\beta\xi_{III}^{2}+\gamma\xi_{III}^{3}$ and
potential $V_{iv}^{III}\left(t,x,y,z\right)$ the corresponding Lie invariants
are
$\left(t,y-\frac{\beta}{\alpha}x,\left(z+\frac{\gamma}{\alpha}\right)e^{-x},u\right)$.

In the case where the proper CKV produces a Lie symmetry, then for the Klein-
Gordon equation (75) with potential function $V_{v}^{III}\left(t,x,y,z\right)$
the Lie invariants are
$\left\\{x,ye^{-\frac{m}{2}t},ze^{-\frac{m\lambda}{2}t},e^{m\lambda
t}u\right\\}$, while for the potential function
$V_{v}^{III}\left(t,x,y,z\right)$ the resulting Lie invariants are
$\left\\{x-\frac{\alpha
m}{2}t,\left(y+\beta\right)e^{-\frac{m}{2}t},\left(z+\frac{\gamma}{\lambda+\alpha}\right)e^{-\frac{m\left(\lambda+\alpha\right)}{2}t},e^{m\lambda
t}u\right\\}$.

####  4.2.2 Conservation laws

For the Noetherian conservation laws for the Klein-Gordon equation (75) it
follows that for $V_{i}^{III}\left(t,x,y,z\right)$ it follows

$I^{x}\left(\xi_{III}^{1}\right)=\frac{NABC}{2}e^{-x}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{1}{B^{2}}u_{,y}^{2}+\frac{e^{2x}}{C^{2}}u_{,z}^{2}\right)-V_{i}^{III}\left(t,x,y,z\right)u^{2}\right)\right)$
(78)
$I^{z}\left(\xi_{III}^{1}\right)=\frac{NABC}{2}e^{-x}\left(z\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{1}{B^{2}}u_{,y}^{2}+\frac{e^{2x}}{C^{2}}u_{,z}^{2}\right)-V_{i}^{III}\left(t,x,y,z\right)u^{2}\right)\right)$
(79) $I^{t}\left(\xi_{III}^{1}\right)=0\text{ and
}I^{y}\left(\xi_{III}^{1}\right)=0\text{.}$ (80)

For the potential function $V_{ii}^{III}\left(t,x,y,z\right)$ the Noetherian
conservation law has the following components

$I^{y}\left(\xi_{III}^{2}\right)=\frac{NABC}{2}e^{-x}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{1}{B^{2}}u_{,y}^{2}+\frac{e^{2x}}{C^{2}}u_{,z}^{2}\right)-V_{ii}^{III}\left(t,x,y,z\right)u^{2}\right)\right)$
(81) $I^{t}\left(\xi_{III}^{2}\right)=0\text{
,~{}}I^{x}\left(\xi_{III}^{2}\right)=0~{}\text{and
}I^{z}\left(\xi_{III}^{2}\right)=0\text{.}$ (82)

Similarly, for $V_{iii}^{III}\left(t,x,y,z\right)$ we determine the
conservation law

$I^{z}\left(\xi_{III}^{3}\right)=\frac{NABC}{2}e^{-x}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{1}{B^{2}}u_{,y}^{2}+\frac{e^{2x}}{C^{2}}u_{,z}^{2}\right)-V_{ii}^{III}\left(t,x,y,z\right)u^{2}\right)\right)$
(83) $I^{t}\left(\xi_{III}^{3}\right)=0\text{
,~{}}I^{x}\left(\xi_{III}^{3}\right)=0~{}\text{and
}I^{y}\left(\xi_{III}^{3}\right)=0\text{.}$ (84)

For the Klein-Gordon equation with potential
$V_{iv}^{III}\left(t,x,y,z\right)$ the conservation law has the components

$I^{x}\left(\alpha\xi_{III}^{1}+\beta\xi_{III}^{2}+\gamma\xi_{III}^{3}\right)=\alpha
I^{x}\left(\xi_{III}^{1}\right)~{},$ (85)
$I^{y}\left(\alpha\xi_{III}^{1}+\beta\xi_{III}^{2}+\gamma\xi_{III}^{3}\right)=\beta
I^{y}\left(\xi_{III}^{2}\right)~{},$ (86)
$I^{z}\left(\alpha\xi_{III}^{1}+\beta\xi_{III}^{2}+\gamma\xi_{III}^{3}\right)=\alpha
I^{z}\left(\xi_{III}^{1}\right)+\gamma I^{z}\left(\xi_{III}^{3}\right),$ (87)

and

$I^{t}\left(\alpha\xi_{III}^{1}+\beta\xi_{III}^{2}+\gamma\xi_{III}^{3}\right)=0,$
(88)

with potential function $V_{iv}^{III}\left(t,x,y,z\right)$.

Moreover, for $V_{v}^{III}\left(t,x,y,z\right)$ the conservation law has the
following components

$I^{t}\left(X_{3}\right)=\frac{\bar{A}^{4}e^{\frac{\left(3m\lambda-1\right)}{2}t}e^{-x}}{2}\left(\frac{2}{m}H_{V}^{III}-\left(\left(\frac{2}{m}\frac{\bar{A}_{,t}}{\bar{A}}+\lambda\right)u\right)\frac{1}{A}u^{,t}-\frac{1}{2\bar{A}e^{\frac{m}{2}\lambda
t}}\left(\frac{2}{m}\frac{\bar{A}_{,t}}{\bar{A}}+\lambda\right)_{t}u^{2}\right).$
(89)
$I^{x}\left(X_{3}\right)=\frac{\bar{A}^{4}e^{\frac{\left(3m\lambda-1\right)}{2}t}e^{-x}}{2}\left(-\left(\left(\frac{2}{m}\frac{\bar{A}_{,t}}{\bar{A}}+\lambda\right)u\right)\frac{1}{\bar{A}e^{\frac{1}{2}\lambda
t}}u^{,x}\right).$ (90)
$I^{y}\left(X_{3}\right)=\frac{\bar{A}^{4}e^{\frac{\left(3m\lambda-1\right)}{2}t}e^{-x}}{2}\left(yH_{V}^{III}-\left(\left(\frac{2}{m}\frac{A_{,t}}{A}+\lambda\right)u\right)\frac{1}{\bar{A}e^{m\left(\lambda-1\right)t}}u^{,y}\right).$
(91)
$I^{z}\left(X_{3}\right)=\frac{\bar{A}^{4}e^{\frac{\left(3m\lambda-1\right)}{2}t}e^{-x}}{2}\left(\lambda
zH_{V}^{III}-\left(\left(\frac{2}{m}\frac{\bar{A}_{,t}}{\bar{A}}+\lambda\right)u\right)\frac{1}{\bar{A}e^{-x}}u^{,z}\right).$
(92)

where now

$H_{V}^{III}=\left(\left(-\frac{1}{\bar{A}^{2}e^{m\lambda
t}}u_{,t}^{2}+\frac{1}{\bar{A}^{2}}u_{,x}^{2}+\frac{1}{\bar{A}^{2}e^{m\left(\lambda-1\right)t}}u_{,y}^{2}+\frac{e^{2x}}{C^{2}}u_{,z}^{2}\right)-V_{v}^{III}\left(t,x,y,z\right)u^{2}\right)$

Finally, for the potential $V_{vi}^{III}\left(t,x,y,z\right)$ the conservation
law for the Klein-Gordon equation (75) related to the generic symmetry vector
$X_{3}+\alpha\xi_{III}^{1}+\beta\xi_{III}^{2}+\gamma\xi_{III}^{3}-\left(\frac{2}{m}\frac{A_{,t}}{A}+\lambda\right)u\partial_{u}$
has the following components

$\displaystyle I^{t}$ $\displaystyle=$ $\displaystyle
I^{t}\left(X_{3}\right)~{},$ (93) $\displaystyle I^{x}$ $\displaystyle=$
$\displaystyle I^{x}\left(X_{3}\right)+\alpha
I^{x}\left(\xi_{III}^{1}\right)~{},$ (94) $\displaystyle I^{y}$
$\displaystyle=$ $\displaystyle I^{y}\left(X_{3}\right)+\beta
I^{y}\left(\xi_{III}^{2}\right)~{},$ (95) $\displaystyle I^{z}$
$\displaystyle=$ $\displaystyle I^{z}\left(X_{3}\right)+\alpha
I^{z}\left(\xi_{III}^{1}\right)+\gamma I^{z}\left(\xi_{III}^{3}\right)~{},$
(96)

with $N=\bar{A}\left(t\right)e^{\frac{m}{2}\lambda t}$,
$B=\bar{A}\left(t\right)e^{\frac{m}{2}\left(\lambda-1\right)t}$ ,
$C\left(t\right)=\bar{A}\left(t\right)$ and $A\left(t\right)=e^{m\lambda
t}\bar{A}\left(t\right)$ and potential function
$V_{vii}^{III}\left(t,x,y,z\right).$

###  4.3 Bianchi V

For the Bianchi V spacetime the Klein-Gordon equation is

$\left(-\frac{u_{,tt}}{N^{2}}+\frac{u_{,xx}}{A^{2}}+e^{-2x}\left(\frac{u_{,yy}}{B^{2}}+\frac{u_{,zz}}{C^{2}}\right)\right)+\frac{1}{N^{2}}\left(\frac{N_{.t}}{N}u_{,t}-\frac{A_{,t}}{A}u_{,x}-\frac{C_{,t}}{C}u_{,y}-\frac{C_{,t}}{C}u_{,z}\right)+\frac{2}{A^{4}}u_{,x}-\frac{V\left(t,x,y,z\right)}{A^{2}B^{2}C}u^{2}=0.$
(97)

The KVs of the Bianchi V spacetime are
$\xi_{V}^{1}=\partial_{x}-y\partial_{z}-z\partial_{z}$,
$\xi_{V}^{2}=\partial_{y}$ and $\xi_{V}^{3}=\partial_{z}$.

Therefore, from Theorem 2 we find that (i) $\xi_{III}^{1}$ is a Lie symmetry
for the Klein-Gordon equation (97) when
$V_{i}^{V}\left(t,x,y,z\right)=V\left(t,e^{x}y,e^{x}z\right)$; (ii)
$\xi_{V}^{2}$ is a Lie symmetry for
$V_{ii}^{V}\left(t,x,y,z\right)=V\left(t,x,z\right)$; (iii) $\xi_{V}^{3}$ is a
Lie symmetry when $V_{iii}^{V}\left(t,x,y,z\right)=V\left(t,x,y\right)$; (iv)
$\alpha\xi_{V}^{1}+\beta\xi_{V}^{2}+\gamma\xi_{V}^{3}$ is a Lie symmetry when
$V_{iv}^{V}\left(t,x,y,z\right)=V\left(t,\left(y-\frac{\beta}{\alpha}\right)e^{x},\left(z-\frac{{}^{\gamma}}{\alpha}\right)e^{x}\right)$.

Finally, for the line element (17) the vector field
$X=X_{3}-\left(\frac{2}{m}\frac{A_{,t}}{A}+\lambda\right)u\partial_{u}$ is a
Lie symmetry for the Klein-Gordon equation when the potential is of the form
of function (76), while
$X=X_{3}+\alpha\xi_{V}^{1}+\beta\xi_{V}^{2}+\gamma\xi_{V}^{3}-\left(\frac{2}{m}\frac{A_{,t}}{A}+\lambda\right)u\partial_{u}$
is a Lie symmetry when

$\displaystyle V_{v}^{V}\left(t,x,y,z\right)$ $\displaystyle=$
$\displaystyle\frac{1}{A^{2}}e^{-m\lambda t}U\left(x-\frac{\alpha
m}{2}t,\left(y+\frac{\beta}{\alpha-1}\right)e^{\frac{m\left(\alpha-1\right)}{2}t},\left(z+\frac{\gamma}{\alpha-\lambda}\right)e^{-\frac{m\left(\alpha-\lambda\right)}{2}t}\right)$
(98)
$\displaystyle~{}~{}+\frac{m\left(\lambda-1\right)A_{,t}+2A_{,tt}}{2A^{3}}.$

####  4.3.1 Invariant functions

As previously, we determine the Lie invariants related to the admitted
symmetry vectors for each potential functional. Indeed, for the potential
$V_{i}^{V}\left(t,x,y,z\right)$ the invariant functions related to the Lie
symmetry $\xi_{III}^{1}$ are $\left\\{t,e^{x}y,e^{x}z,u\right\\}$. For the
potential function $V_{ii}^{V}\left(t,x,y,z\right)$ we determine the
invariants $\left\\{t,x,z,u\right\\}$ while for
$V_{iii}^{V}\left(t,x,y,z\right)$ the invariants are
$\left\\{t,x,y,u\right\\}$. Moreover, for $V_{iv}^{V}\left(t,x,y,z\right)$ the
corresponding invariant functions related to the generic vector field
$\alpha\xi_{V}^{1}+\beta\xi_{V}^{2}+\gamma\xi_{V}^{3}$ are
$\left\\{t,\left(y-\frac{\beta}{\alpha}\right)e^{x},\left(z-\frac{{}^{\gamma}}{\alpha}\right)e^{x},u\right\\}$.

Finally, for the remaining cases where the proper CKV generates Lie
symmetries, it follows that for potential $V_{v}^{III}\left(t,x,y,z\right)$
the Lie invariants are
$\left\\{x,ye^{-\frac{m}{2}t},ze^{-\frac{m\lambda}{2}t},e^{m\lambda
t}u\right\\}$ while for the potential $V_{v}^{V}\left(t,x,y,z\right)$ and the
Lie symmetry
$X_{3}+\alpha\xi_{V}^{1}+\beta\xi_{V}^{2}+\gamma\xi_{V}^{3}-\left(\frac{2}{m}\frac{A_{,t}}{A}+\lambda\right)u\partial_{u}$
the corresponding Lie invariants are $\left\\{x-\frac{\alpha
m}{2}t,\left(y+\frac{\beta}{\alpha-1}\right)e^{\frac{m\left(\alpha-1\right)}{2}t},\left(z+\frac{\gamma}{\alpha-\lambda}\right)e^{-\frac{m\left(\alpha-\lambda\right)}{2}t},e^{m\lambda
t}u\right\\}$.

We proceed with the derivation of the conservation laws.

####  4.3.2 Conservation laws

The conservation law for the Klein-Gordon equation (97) and potential function
$V_{i}^{V}\left(t,x,y,z\right)$ has the following components

$I^{x}\left(\xi_{V}^{1}\right)=\frac{NABC}{2}e^{x}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{e^{-x}}{B^{2}}u_{,y}^{2}+\frac{e^{-x}}{C^{2}}u_{,z}^{2}\right)-V_{i}^{V}\left(t,x,y,z\right)u^{2}\right)\right),$
(99)
$I^{y}\left(\xi_{V}^{1}\right)=\frac{NABC}{2}e^{x}\left(-y\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{e^{-x}}{B^{2}}u_{,y}^{2}+\frac{e^{-x}}{C^{2}}u_{,z}^{2}\right)-V_{i}^{V}\left(t,x,y,z\right)u^{2}\right)\right),$
(100)
$I^{z}\left(\xi_{V}^{1}\right)=\frac{NABC}{2}e^{x}\left(-z\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{e^{-x}}{B^{2}}u_{,y}^{2}+\frac{e^{-x}}{C^{2}}u_{,z}^{2}\right)-V_{i}^{V}\left(t,x,y,z\right)u^{2}\right)\right),$
(101)

and

$I^{t}\left(\xi_{V}^{1}\right)=0\text{ .}$ (102)

For $V_{ii}^{V}\left(t,x,y,z\right)$ it follows

$I^{y}\left(\xi_{V}^{2}\right)=\frac{NABC}{2}e^{x}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{e^{-x}}{B^{2}}u_{,y}^{2}+\frac{e^{-x}}{C^{2}}u_{,z}^{2}\right)-V_{ii}^{V}\left(t,x,y,z\right)u^{2}\right)\right),$
(103) $I^{t}\left(\xi_{V}^{2}\right)=0\text{
,~{}}I^{x}\left(\xi_{V}^{2}\right)=0\text{ and
}I^{z}\left(\xi_{V}^{2}\right)=0\text{ }.$ (104)

For $V_{iii}^{V}\left(t,x,y,z\right)$ we find

$I^{z}\left(\xi_{V}^{3}\right)=\frac{NABC}{2}e^{x}\left(\left(\left(-\frac{1}{N^{2}}u_{,t}^{2}+\frac{1}{A^{2}}u_{,x}^{2}+\frac{e^{-x}}{B^{2}}u_{,y}^{2}+\frac{e^{-x}}{C^{2}}u_{,z}^{2}\right)-V_{iii}^{V}\left(t,x,y,z\right)u^{2}\right)\right),$
(105) $I^{t}\left(\xi_{V}^{3}\right)=0\text{
,~{}}I^{x}\left(\xi_{V}^{3}\right)=0\text{ and
}I^{y}\left(\xi_{V}^{3}\right)=0\text{ }.$ (106)

For the generic vector field
$\alpha\xi_{V}^{1}+\beta\xi_{V}^{2}+\gamma\xi_{V}^{3}$ and potential
$V_{iv}^{V}\left(t,x,y,z\right)$ it follows

$I^{x}\left(\alpha\xi_{V}^{1}+\beta\xi_{V}^{2}+\gamma\xi_{V}^{3}\right)=\alpha
I^{x}\left(\xi_{V}^{1}\right)$
$I^{y}\left(\alpha\xi_{V}^{1}+\beta\xi_{V}^{2}+\gamma\xi_{V}^{3}\right)=\alpha
I^{y}\left(\xi_{V}^{1}\right)+\beta I^{y}\left(\xi_{V}^{2}\right)$
$I^{z}\left(\alpha\xi_{V}^{1}+\beta\xi_{V}^{2}+\gamma\xi_{V}^{3}\right)=\alpha
I^{z}\left(\xi_{V}^{1}\right)+\gamma I^{z}\left(\xi_{V}^{3}\right)$

and

$I^{t}\left(\alpha\xi_{V}^{1}+\beta\xi_{V}^{2}+\gamma\xi_{V}^{3}\right)=0,$

with potential function $V_{iv}^{V}\left(t,x,y,z\right).$

From the Lie symmetry vector
$X=X_{3}-\left(\frac{2}{m}\frac{A_{,t}}{A}+\lambda\right)u\partial_{u}$ we
determine the conservation law

$I^{t}\left(X_{3}\right)=\frac{\bar{A}^{4}e^{\frac{\left(3m\lambda-1\right)}{2}t}e^{x}}{2}\left(\frac{2}{m}H_{V}^{V}-\left(\left(\frac{2}{m}\frac{\bar{A}_{,t}}{\bar{A}}+\lambda\right)u\right)\frac{1}{A}u^{,t}-\frac{1}{2\bar{A}e^{\frac{m}{2}\lambda
t}}\left(\frac{2}{m}\frac{\bar{A}_{,t}}{\bar{A}}+\lambda\right)_{t}u^{2}\right).$
(107)
$I^{x}\left(X_{3}\right)=\frac{\bar{A}^{4}e^{\frac{\left(3m\lambda-1\right)}{2}t}e^{x}}{2}\left(-\left(\left(\frac{2}{m}\frac{\bar{A}_{,t}}{\bar{A}}+\lambda\right)u\right)\frac{1}{\bar{A}e^{\frac{1}{2}\lambda
t}}u^{,x}\right).$ (108)
$I^{y}\left(X_{3}\right)=\frac{\bar{A}^{4}e^{\frac{\left(3m\lambda-1\right)}{2}t}e^{x}}{2}\left(yH_{V}^{V}-\left(\left(\frac{2}{m}\frac{A_{,t}}{A}+\lambda\right)u\right)\frac{1}{\bar{A}e^{m\left(\lambda-1\right)t}e^{x}}u^{,y}\right).$
(109)
$I^{z}\left(X_{3}\right)=\frac{\bar{A}^{4}e^{\frac{\left(3m\lambda-1\right)}{2}t}e^{x}}{2}\left(\lambda
zH_{V}^{V}-\left(\left(\frac{2}{m}\frac{\bar{A}_{,t}}{\bar{A}}+\lambda\right)u\right)\frac{1}{\bar{A}e^{x}}u^{,z}\right).$
(110)

in which

$H_{V}^{V}=\left(\left(-\frac{1}{\bar{A}^{2}e^{m\lambda
t}}u_{,t}^{2}+\frac{1}{\bar{A}^{2}}u_{,x}^{2}+\frac{e^{-x}}{\bar{A}^{2}e^{m\left(\lambda-1\right)t}}u_{,y}^{2}+\frac{e^{-x}}{C^{2}}u_{,z}^{2}\right)-V_{v}^{III}\left(t,x,y,z\right)u^{2}\right).$

Finally, the Klein-Gordon equation in the Bianchi V background space with
potential function$~{}V_{v}^{V}\left(t,x,y,z\right)$ admits the conservation
law with components

$\displaystyle I^{t}$ $\displaystyle=$ $\displaystyle
I^{t}\left(X_{3}\right)~{},$ (111) $\displaystyle I^{x}$ $\displaystyle=$
$\displaystyle I^{x}\left(X_{3}\right)+\alpha
I^{x}\left(\xi_{V}^{1}\right)~{},$ (112) $\displaystyle I^{y}$
$\displaystyle=$ $\displaystyle I^{y}\left(X_{3}\right)+\alpha
I^{y}\left(\xi_{V}^{1}\right)+\beta I^{y}\left(\xi_{V}^{2}\right)~{},$ (113)
$\displaystyle I^{z}$ $\displaystyle=$ $\displaystyle
I^{z}\left(X_{3}\right)+\alpha I^{z}\left(\xi_{V}^{1}\right)+\gamma
I^{z}\left(\xi_{V}^{3}\right)~{}.$ (114)

##  5 Conclusions

We performed a detailed study for infinitesimal transformations which leave
invariant the Klein-Gordon equation with a non-constant potential function in
curved spacetimes. Specifically, we determined all the admitted Lie and
Noether symmetries for the Klein-Gordon equation. We considered four-
dimensional Riemannian manifolds which describe homogeneous and anisotropic
cosmologies. We wrote the Klein-Gordon equation in the case of Bianchi I,
Bianchi III and Bianchi V spacetimes and we determined all the unknown
functional forms of the potential function where the Klein-Gordon equations
admit non-trivial Lie and Noether symmetries.

We made use of some previous results which relate the infinitesimal
transformations, i.e. the Lie and Noether symmetries, for the Klein-Gordon
equation to the elements of the conformal algebra for the metric tensor of the
Riemannian manifold where the Laplace operator is defined. Thus, we performed
a detailed presentation of the CKVs for the three spacetimes of our
consideration. These spacetimes for arbitrary scale factors have a three-
dimensional conformal algebra which consists of these KVs. However, for
special functions of the scale factors the spacetimes admit a proper CKV.
There are two forms for the line-element of Bianchi I spacetime where a proper
CKV exists, and there is one specific form for the line element of Bianchi III
and one specific line element for the Bianchi V spacetime where one proper CKV
exist.

Thus, for all the specific line elements we present in a systematic way all
the functional forms for the potential for the Klein-Gordon equation where Lie
and Noether symmetries exist. Such an analysis is important in order to
understand the relation of symmetries of differential equations with the
background geometry, as it also shows how symmetries can be derived in a
simple and systematic approach by using tools from differential geometry. Last
but not least, the Noetherian conservation laws can be easily constructed with
the application of Noether’s second theorem.

Acknowledgments: The author thanks for the support of Vicerrectoría de
Investigación y Desarrollo Tecnológico (Vridt) at Universidad Católica del
Norte through Núcleo de Investigación Geometría Diferencial y Aplicaciones,
Resolución Vridt No - 096/2022.

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11institutetext: Indian Institute of Technology, Kanpur, India
11email<EMAIL_ADDRESS>
22institutetext: Vellore Institute of Technology, Chennai, India
22email<EMAIL_ADDRESS>
# nigam@COLIEE-22: Legal Case Retrieval and Entailment using Cascading of
Lexical and Semantic-based models

Shubham Kumar Nigam∗ 11 Navansh Goel∗ 22

###### Abstract

This paper describes our submission to the Competition on Legal Information
Extraction/Entailment 2022 (COLIEE-2022) workshop on case law competition for
tasks 1 and 2. Task 1 is a legal case retrieval task, which involves reading a
new case and extracting supporting cases from the provided case law corpus to
support the decision. Task 2 is the legal case entailment task, which involves
the identification of a paragraph from existing cases that entails the
decision in a relevant case. We employed the neural models Sentence-BERT and
Sent2Vec for semantic understanding and the traditional retrieval model BM25
for exact matching in both tasks. As a result, our team ("nigam") ranked 5th
among all the teams in Tasks 1 and 2. Experimental results indicate that the
traditional retrieval model BM25 still outperforms neural network-based
models.

###### Keywords:

BM25 Sentence-BERT Sent2vec Reduced-Space.

**footnotetext: These authors contributed equally to this work

##  1 Introduction

Many countries, such as India, the United States, Canada, Australia, and
Africa, follow the case law system. These case law systems are legal systems
that give great weight to judicial precedent, and the volume of information
produced in the legal sector nowadays is overwhelming. However, it takes
significant efforts of legal practitioners to search for relevant cases from
the database and extract the entailment parts manually with the rapid growth
of digitalized legal documents. Therefore, an efficient legal assistant system
to alleviate the heavy document work is necessary for Legal-AI.

The Competition on Legal Information Extraction and Entailment (COLIEE)
workshop has been organized for several years to assist the legal research
community. They provide the legal texts and specific problems in the legal
domain such as question-answering, case law retrieval, case law entailment,
statute law retrieval, and statute law entailment to help the research
community develop Legal-AI techniques.

This paper introduces our approaches to completing two case law tasks, Task 1
(the retrieval task) and Task 2 (the entailment task). We employed both
traditional retrieval models like BM25[6] for exact matching and semantic
understanding like Sentence-BERT[5], Sent2Vec[4] in both task, and their
combination in Task 1. In combination, we first select top-K candidates
according to the BM25 rankings and afterward get the representation features
of each sentence for the document we used pre-trained Sentence-BERT and
Sent2Vec via comparing paragraph-level interaction. Furthermore, we used
cosine similarity with the max-pooling strategy to get the final document
score between query and noticed cases. As a result, our team ranked 5th among
all the teams in both tasks. Experimental results suggest that the exact
matching using the traditional retrieval model BM25 still outperforms
representation features generated by neural network-based models. We released
the codes for all subtasks via
GitHub111https://github.com/ShubhamKumarNigam/COLIEE-22.

##  2 The Task

###  2.1 Task 1 - Legal Case Retrieval

The main objective of legal case law retrieval is to obtain relevant
supporting documents for a given query case $Q$. The task involves retrieving
noticed cases $N=\\{N1,N2,...,N_{m}\\}\mid N\subseteq S$ given that
$S=\\{S1,S2,S3,...,S_{n}\\}$ belonging to a given case law corpus of all
possible supporting cases. The corpus predominantly consists of case law
documents from the Federal Court of Canada database provided by Compass Law.

The query case documents contain suppressed reference markers instead of
actual references to precedent case laws, compelling the participants to
design models that can understand the text around the references and return
valid noticed cases. A supporting case $S_{k}:S_{k}\in S$, is a noticed case
for query $Q_{j}:Q_{j}\in Q$, _iff_ $Q_{j}$ contains a reference to $S_{k}$.

###  2.2 Task 2 - Legal Case Entailment

Task 2 comprises identifying a specific paragraph from a given supporting case
that entails the decision for the query case. Thus, given a query case $Q$, a
supporting case $R$ and paragraphs $P=\\{P1,P2,P3,....,Pn\\}$ such that
$P\subseteq R$, the task requires one to correctly identify $P_{i}\in P$ such
that it entails the decision for query case $Q$.

##  3 Data Corpus

The data corpus for both Task 1 and Task 2 belongs to a database of case law
documents from the Federal Court of Canada provided by Compass Law. Table 1
presents the dataset statistics for both the tasks. For Task 1, 898 query
cases were given against 3531 candidate cases as a part of the training data
corpus. The test dataset had 300 query cases against 1263 candidate cases. For
Task 2, 525 query cases were provided for training against 18,740 paragraphs.
There were 100 query cases against 3278 candidate paragraphs as part of the
testing dataset. The organizers provided the candidate paragraphs for a given
query case separately. On average, there are 35.627 candidate paragraphs for
each query case in the training dataset and 32.455 candidate paragraphs for
each query case in the testing dataset.

On further analysis, we found that the average number of relevant candidate
cases for Task 1 was 4.684 cases for the training dataset and 4.21 relevant
cases for the test dataset. As a part of our methodology, we predict the top 5
possible noticed cases for each query case based on these numbers. The average
number of relevant paragraphs for Task 2 was a little over 1 paragraph, 1.14
paragraphs for training and 1.18 paragraphs for testing. These numbers show
that most documents for Task 2 had their decisions entailed inside a single
paragraph. The table also shows the average length of each document. The
average number of words is higher in the testing query dataset than in the
training query dataset for Task 1.

On the contrary, the average number of words per candidate document decreases
from the training data to the testing data. The opposite trend is observed for
Task 2, where the average number of words per query document decreases from
training data to testing data. In contrast, the average number of words per
candidate paragraph increases from training to testing.

Table 1: Statistics of the training and test data for Task1 and Task2

| Task 1 | Task 2
---|---|---
| Train | Test | Train | Test
# of queries | 898 | 300 | 525 | 100
# of candidate cases/paragraphs | 3,531 | 1,263 | 18,740 | 3,278
avg # of candidate paragraphs per query | - | - | 35.627 | 32.455
avg # of relevant candidates/paragraphs | 4.684 | 4.210 | 1.14 | 1.18
avg query length (words) | 28,866.73 | 32,461.27 | 24,966.55 | 22,326.95
avg candidate length (words) | 30,113.09 | 29,756.87 | 636.08 | 643.69

###  3.1 Preprocessing

Since the given documents are very long and represent the entire document
singly is not efficient. Also, it is often noted that case judgments cite
multiple cases and write multiple sentences about them. So we segment the
document into meaningful sentences using the spacy
library222https://spacy.io/usage/linguistic-features#sbd and retrieve relevant
contents using the citation marker "FRAGMENT_SUPPRESSED," which can say
context around the citation. These sentences feed the models to make a corpus
or generate distributed representations.

###  3.2 Evaluation Metrics

The evaluation metrics of Tasks 1 and 2 are precision, recall, and F-measure.
All the metrics are micro-average, which means the evaluation measure is
calculated using the results of all queries. Definition of these measures are
as follows:

$Precision=\frac{\text{\\# correctly retrieved cases(paragraphs) for all
queries}}{\text{\\# retrieved cases(paragraphs) for all queries}}$ (1)

$Recall=\frac{\text{\\# correctly retrieved cases(paragraphs) for all
queries}}{\text{\\# relevant cases(paragraphs) for all queries}}$ (2)

$F-measure=\frac{\text{2 * Precision * Recall}}{\text{Precision + Recall}}$
(3)

##  4 Our Methods

###  4.1 BM25

In previous COILEE submissions, many participants Kim et al. [2], Ma et al.
[3], Rosa et al. [7], Shao et al. [8] and Shao et al. [9] in IJCAI-20
concluded that traditional bag-of-words IR models have competitive
performances in legal case retrieval tasks. So our first preference is to try
the exact matching-based BM25 model [6], which is a probabilistic relevance
model based on a bag-of-words approach. The score of a document $D$ given a
query $Q$ which contains the words $q_{1},...,q_{n}$ is given by:

$\text{score}(D,Q)=\sum_{i=1}^{n}\text{IDF}(q_{i})\cdot\frac{f(q_{i},D)\cdot(k_{1}+1)}{f(q_{i},D)+k_{1}\cdot(1-b+b\cdot\frac{|D|}{\text{avgdl}})}$
(4)

where $f(q_{i},D)$ is $q_{i}$’s term frequency in the document $D$, $|D|$ is
the length of the document $D$ in words, and avgdl is the average document
length in the text collection from which documents are drawn. $k_{1}$ and $b$
are free parameters.

$\text{IDF}(q_{i})=ln(\frac{N-n(q_{i})+0.5}{n(q_{i})+0.5}+1)$ (5)

where $N$ is the total number of documents in the collection, and $n(q_{i})$
is the number of documents containing
$q_{i}$333https://en.wikipedia.org/wiki/Okapi_BM25.

We tried BM25 using Rank-BM25
library444https://github.com/dorianbrown/rank_bm25, where the algorithms are
taken from the paper Trotman et al. [10]. Although we are not getting good
results, we tried to implement Okapi BM25 with sklearn’s
TfidfVectorizer555https://scikit-
learn.org/stable/modules/generated/sklearn.feature_extraction.text.TfidfVectorizer.html
which converts a collection of raw documents to a matrix of TF-IDF features.
TfidfVectorizer has valuable features; some important feature descriptions
are:

  * •

stop_words: Either we can pass string ‘english’ or a list containing stop
words, all of which will be removed from the resulting tokens.

  * •

ngram_range: The lower and upper boundary of the range of n-values for
different n-grams to be extracted. For example, an ngram_range of (1, 1) means
only unigrams, and (1, 2) means unigrams and bigrams.

  * •

max_df: While building the vocabulary, ignore terms with a document frequency
strictly higher than the given threshold (corpus-specific stop words). The
range is [0.0, 1.0].

  * •

min_df: While building the vocabulary, ignore terms with a document frequency
strictly lower than the given threshold. This value is also called a cut-off
in the literature. The range is [0.0, 1.0].

  * •

norm: Each output row will have a unit norm, either:

    * –

‘l2’: The sum of squares of vector elements is 1. The cosine similarity
between two vectors is their dot product when the l2 norm has been applied.

    * –

‘l1’: Sum of absolute values of vector elements is 1.

These features are beneficial for BM25 to extract better features, and even we
can restrain these features. Our results depict that after employing
TfidfVectorizer, BM25 performed much better.

###  4.2 Sent2Vec

Sent2Vec is an Unsupervised Learning of Sentence Embeddings or distributed
representations of sentences using Compositional N-Gram Features. It can be
thought of as an extension of word2vec (CBOW) to sentences. Sentence embedding
is the average of the source word embeddings of its constituent words. We
generate features from the pre-trained model "sent2vec_wiki_bigrams" from
Sent2vec666https://github.com/epfml/sent2vec library and paper Pagliardini et
al. [4], which is trained on the English Wikipedia dataset and embedding
dimensions are 700. Now to find the most similar sentences between query case
and candidate case in a dataset, we compared using cosine-
similarity777https://scikit-
learn.org/stable/modules/generated/sklearn.metrics.pairwise.cosine_similarity.html
using max-pool strategy. Finally, we used the max-pooling strategy between
query and noticed cases to get the most similar document.

###  4.3 Sentence-BERT

Since we need sentence/paragraph representation, instead of using any
transformer and generating sentence embedding using a pooling strategy, it
will be more suitable to directly use a model to derive semantically
meaningful representations for sentences. We use the Sentence-
BERT888https://www.sbert.net/ framework based on the paper [5], which provides
an easy method to compute dense vector representations for sentences and
paragraphs. This model is based on transformer networks like BERT/RoBERTa, but
this modification of the pre-trained BERT network uses siamese and triplet
network structures to derive semantically meaningful sentence embeddings. We
generate dense vectors from the pre-trained model "all-mpnet-base-v2" from
Sentence-BERT999https://www.sbert.net/docs/pretrained_models.html library,
which is trained on a large and diverse dataset of over 1 billion training
pairs, max sequence length 384, mean-pooling, and generates 768 dimensions. To
find the most similar paragraphs between query and candidate cases in a
dataset, we compared using cosine-similarity using the max-pool strategy.
Finally, we used the max-pooling strategy between query and noticed cases to
get the most similar document.

###  4.4 Reduced-Space

As illustrated in Figure 1, we first represent the query corpus in paragraphs
$Q=(Q_{p1},Q_{p2},...,Q_{pN})$ by extracting the sentences which contain the
"FRAGMENT_SUPPRESSED" marker say $i$, three previous $i-3$ and three following
sentences $i+3$ and citation corpus by segmentation of logical paragraphs
$C=(C_{p1},C_{p2},...,C_{pM})$. In total, we obtain $N*M$ pairs of paragraphs.
Then, we used a lexical model to calculate the similarity between a given
query and the entire case law corpus on paragraph-wise segmented documents. We
limit the searching space from 3531 to 100 candidate cases by picking the top
100 cases with the highest BM25 scores for a given query. After reducing the
searching space to 100 documents, we want to extract the semantic relationship
between the query and candidate paragraphs pairs, e.g., $(Q_{pi};C_{pj})$. To
obtain this relationship score, we use the pre-trained model Sentence-BERT and
Sent2vec. We used the max-pooling strategy to find the most similar paragraphs
and documents in the same approach as above.

Figure 1: An illustration of Reduced-Space Architecture.

##  5 Results and Analysis

###  5.1 Task-1

We submitted three official runs in table 2, run-1 is BM25, in a cascaded
manner reduced space after using BM25 Recall@100, run-2 is pre-trained
Sentence-BERT, and run-3 is pre-trained Sent2vec. All these official runs have
done experiments on the paragraph level comparison, and every query case
predicts a top-5 citation for both training and test dataset because the
average number of relevant cases is approximately 5 in training data. We get
the top score in using the traditional model BM25 for exact matching; on the
other hand, in semantic understanding models, Sentence-BERT gets second, and
Sent2vec gets third. Our team ("nigam") ranked 5th among all the teams in
Tasks 1, and the results indicate that the traditional retrieval model BM25
still outperforms neural network-based models.

Apart from that, we tried several experiments without reduced space means the
search space was the entire citation corpus in Sent2vec and Sentence-BERT.
Similar behavior can be seen in table 3 for the training dataset, where BM25
performs better than Sentence-BERT and Sent2vec in reduced space. Along with
experiments on the paragraph level comparison, we also tried document level
comparison, but they did not give good results. We even included results
without reduced space, and it can easily be observed that the reduced space
results are slightly improved. Training and testing for every query case
predict top-5 case law because the average number of relevant cases is
approximately 5. We remove stop-words, and set parameter values max_df=0.90,
min_df=1, b=0.99, k1=1.6, ngram_range=(2,6). Moreover, other parameters are
set to default for all experiments in BM25.

Table 2: Results on the test set of Task 1

Team | F1 Score | Precision | Recall
---|---|---|---
UA | 0.3715 | 0.4111 | 0.3389
UA | 0.371 | 0.4967 | 0.2961
siat | 0.3691 | 0.3005 | 0.4782
siat | 0.368 | 0.3026 | 0.4695
UA | 0.3559 | 0.363 | 0.3492
siat | 0.2964 | 0.2522 | 0.3595
LeiBi | 0.2923 | 0.3 | 0.285
LeiBi | 0.2917 | 0.2687 | 0.3191
JNLP | 0.2813 | 0.3211 | 0.2502
nigam | 0.2809 | 0.2587 | 0.3072
JNLP | 0.2781 | 0.3144 | 0.2494
DSSR | 0.2657 | 0.2447 | 0.2906
JNLP | 0.2639 | 0.2446 | 0.2866
DSSR | 0.2461 | 0.2267 | 0.2692
TUWBR | 0.2367 | 0.1895 | 0.3151
LeiBi | 0.2306 | 0.2367 | 0.2249
TUWBR | 0.2206 | 0.1683 | 0.3199
nigam | 0.1542 | 0.142 | 0.1686
nigam | 0.1484 | 0.1367 | 0.1623
DSSR | 0.1317 | 0.1213 | 0.1441

Table 3: Task 1 results in the training set. Every query case predicts top-5
case law because the average number of relevant cases is approximately 5. We
remove stop-words, max_df=0.90, min_df=1, b=0.99, k1=1.6, ngram_range=(2,6) in
BM25. Moreover, other parameters are set to default for all experiments.

Model | F1 Score | Precision | Recall
---|---|---|---
BM25 | 0.1957 | 0.1895 | 0.2023
Sent2vec | 0.0812 | 0.0786 | 0.0839
SBERT | 0.0844 | 0.0817 | 0.0873
| BM25 + Sent2vec
---
(Reduced Space)
0.1080 | 0.0961 | 0.1234
| BM25 + SBERT
---
(Reduced Space)
0.1017 | 0.0984 | 0.1051

###  5.2 Task-2

We submitted two official runs in table 4; both runs are based on the BM25
model. Run-1 uses only entailed fragment text as a query, and run-2 uses the
filtered base case such that search entailed fragment text into the base case
then input to the model as searched results along with previous and following
sentences to capture the other relevant information. Every query case predicts
one case law paragraph because the average number of relevant paragraphs is
approximately 1.14 in the training dataset. Our team ("nigam") ranked 5th
among all the teams in Tasks 2, and the results indicate that using only
fragment text as the query will give better scores. Although, we got the
highest Recall score among all participants.

Similar behavior can be seen in table 5 for the training dataset, where BM25
performs better when only fragment text as the query and outputs only a single
paragraph. Apart from that, we also tried two paragraphs prediction for every
query; indeed, that gives a better Recall value, but on the other hand,
precision values drop significantly. We also tried experiments where along
with fragment text, we retrieved previous and following sentences from the
base case after matching fragment text to capture the other relevant
information. For example, $[-i,+j]$ means $i$ previous and $j$ following
sentences from the matched fragment sentences. However, the results show that
results drop as we include the other information. We kept stop-words, and set
parameter values max_df=0.65, min_df=1, b=0.7, k1=1.6, ngram_range=(1,1).
Moreover, other parameters are set to default for all experiments in BM25.

Table 4: Results on the test set of Task 2

Team | F1 Score | Precision | Recall
---|---|---|---
NM | 0.6783 | 0.6964 | 0.661
NM | 0.6757 | 0.7212 | 0.6356
JNLP | 0.6694 | 0.6532 | 0.6864
JNLP | 0.6612 | 0.6452 | 0.678
jljy | 0.6514 | 0.71 | 0.6017
jljy | 0.6514 | 0.71 | 0.6017
JNLP | 0.6452 | 0.6154 | 0.678
jljy | 0.633 | 0.69 | 0.5847
NM | 0.6325 | 0.6379 | 0.6271
UA | 0.5446 | 0.6105 | 0.4915
UA | 0.5363 | 0.7869 | 0.4068
UA | 0.4121 | 0.3049 | 0.6356
nigam | 0.3204 | 0.198 | 0.839
nigam | 0.2104 | 0.13 | 0.5508

Table 5: Task 2 results in the training set. Every query case predicts either
1 or 2 case law paragraphs because the average number of relevant paragraphs
is approximately 1.14. We kept stop-words, max_df=0.65, min_df=1, b=0.7,
k1=1.6, ngram_range=(1,1). Moreover, other parameters are set to default for
all experiments in BM25. Query cases are input such that either only entailed
fragment text or search that text into the base case with previous and
following sentences.

Query case Input |  | # Prediction
---
per query
F1 Score | Precision | Recall
| 1 | 0.6228 | 0.6667 | 0.5843
consider only Entailed Fragment text | 2 | 0.5106 | 0.4010 | 0.7028
| 1 | 0.4359 | 0.4667 | 0.4090
using base case [-1,+1] | 2 | 0.4172 | 0.3276 | 0.5743
| 1 | 0.4110 | 0.4400 | 0.3856
using base case [-1,+2] | 2 | 0.3857 | 0.3029 | 0.5309
| 1 | 0.3790 | 0.4057 | 0.3556
using base case [-1,+3] | 2 | 0.3663 | 0.2876 | 0.5042
| 1 | 0.3594 | 0.3848 | 0.3372
using base case [-2,+3] | 2 | 0.3469 | 0.2724 | 0.4775
| 1 | 0.3488 | 0.3733 | 0.3272
using base case [-3,+3] | 2 | 0.3457 | 0.2714 | 0.4758

##  6 Conclusion

We participated at COLIEE 2022 in Task 1 and Task 2, which allowed exploring
information retrieval challenges in the legal case law retrieval and legal
entailment. Our main objective was to combine traditional lexical retrieval
models BM25 with dense passage retrieval models Sentence-BERT and Sent2vec for
re-ranking the results. We show that the paragraph-level retrieval in the
first stage outperforms the document-level retrieval. Also, overall ranking
after combining lexical and semantic models improves the ranking of dense
passage retrieval models alone for task-1. Although usually, neural network-
based models perform best in the NLP tasks, especially transformers.
Nevertheless, our results and previous COLIEE participants show that these
pre-trained models do not work well in a specialized domain; traditional
lexical retrieval models still perform better.

Furthermore, in task-2, using only fragment text as the query will give better
scores, whereas accommodating other information from the base case only
impairs the accuracy. Since we got the highest Recall score among all
participants, we should also try the reduced space approach. In the future, we
plan to investigate dense retrieval models domain-specific contextualized
language models like LegalBERT [1] and graph neural networks [11].

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