$$\begin{aligned}
\Delta_{1}N &  =g_{ij}N^{,i}N^{,j}\label{CLN.04.2}\\
\Delta_{2}N &  =g_{ij}N^{;ij}.\label{CLN.04.3}$$ From the above discussion it becomes clear that all two dimensional spaces are Einstein Spaces $\left(  \text{i.e. }R_{ab}=\frac{R}{2}g_{ab}\right)  $ and conformally flat. The metric of a two dimensional space can be written in the generic form: $$ds^{2}=N^{2}\left(  x,y\right)  \left(  \varepsilon dx^{2}+dy^{2}\right)
~~,~\varepsilon=\pm1\;.\label{CLN.10}$$

Conformal Lagrangians
---------------------

Due to the fact that almost every dynamical system is described by a corresponding Lagrangian, below we study generically, as much as possible, the problem of the conformal Lagrangians and then we apply the current ideas to the scalar field cosmology. To begin with, consider the Lagrangian of a particle moving under the action of a potential $V(x^{k})$ in a Riemannian space with metric $g_{ij}$ $$L=\frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}-V\left(  x^{k}\right)  ~,~\dot
{x}^{i}=\frac{dx^{i}}{dt}\label{CLN.05}$$ where $t$ is a path parameter. The equations of motion follow from the action $$S=\int dxdtL\left(  x^{k},\dot{x}^{k}\right)  =\int dxdt\left[  \frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}-V\left(  x^{k}\right)  \right]
.\label{CLN.06}$$ Changing the variables in Eq.(\[CLN.06\]) from $t$ to $\tau$ via the relation: $$d\tau=N^{2}\left(  x^{i}\right)  dt \label{tran1A}$$ the action is given by $$S=\int dx\frac{d\tau}{N^{2}\left(  x^{k}\right)  }\left[  \frac{1}{2}g_{ij}N^{4}\left(  x^{k}\right)  x^{\prime i}x^{\prime j}-V\left(
x^{k}\right)  \right]  ~~\label{CLN.07}$$ where $x^{\prime i}=\frac{dx^{i}}{d\tau}$. Obviously, the Lagrangian in the new coordinate system $(\tau,x^{i})$ becomes:$$\bar{L}\left(  x^{k},x^{\prime k}\right)  =\frac{1}{2}N^{2}\left(
x^{k}\right)  g_{ij}x^{\prime i}x^{\prime j}-\frac{V\left(  x^{k}\right)
}{N^{2}\left(  x^{k}\right)  