two metrics $g_{ij},\bar{g}_{ij}$ are said to be conformal or conformally related if there exists a function $N^{2}\left(  x^{k}\right)  $ so that $\bar{g}_{ij}=N^{2}\left(  x^{k}\right)  g_{ij}$. From the mathematical point of view the CKVs form the so called *conformal algebra* of the metric. The conformal algebra contains two closed sub-algebras the *Homothetic algebra* and the *Killing algebra*. Interestingly the above algebras are related as follows: $$KV_{s}\subseteq HV_{s}\subseteq CKV_{s}\;.\label{CLN.02}$$ The dimension of the conformal algebra of an $n-$ dimensional metric $(n>2)$ of constant curvature equals $\frac{(n+1)(n+2)}{2}$, the dimension of the Killing algebra $\frac{n(n+1)}{2}$ and that of the homothetic algebra $\frac{n(n+1)}{2}+1$. Note that two conformally related metrics have the same conformal algebra [@TsampC], however not the same subalgebras. Indeed if $X$ is a CKV for the metric $g_{ij}$ i.e. $\mathcal{L}_{X}g_{ij}=2\psi\left(
x^{k}\right)  g_{ij}$ then for the metric $\bar{g}_{ij}$ the vector $X$ is again a CKV with conformal factor $\bar{\psi}\left(  x^{k}\right)  ,$ that is: $$\mathcal{L}_{X}\bar{g}_{ij}=2\bar{\psi}\left(  x^{k}\right)  \bar{g}_{ij}\label{CLN.03}$$ where the conformal factors $\psi\left(  x^{k}\right)  $, $\bar{\psi}\left(
x^{k}\right)  $ are related as follows: $$\bar{\psi}\left(  x^{k}\right)  =\psi\left(  x^{k}\right)  +\mathcal{L}_{X}\left(  \ln N\right)  .\label{CLN.04}$$

The Ricci scalars of the conformally related metrics $g_{ij},\bar{g}_{ij}$ are related as follows [@HawkingB]:$$\bar{R}=N^{-2}R-2(n-1)N^{-3}\Delta_{2}N-(n-1)(n-4)\Delta_{1}N\label{CLN.04.1}$$ where (note that $\Delta_{2}N$ contains the covariant derivative whereas $\Delta_{1}N$ the partial derivative): 