with the above specified orientation and time-orientation, is called the [*$n$-dimensional Einstein universe*]{}.

In order to describe the conformal geometry of $\mathcal{E}^{1,n-1}$ it is convenient to consider in $\R^{n+2}$ the coordinates $$y_0=\frac{1}{\sqrt{2}}(x_0+x_{n+1}),\quad y_1=x_1, \dots , y_n=x_n,\quad y_{n+1}=\frac{1}{\sqrt{2}}(-x_0+x_{n+1}).$$ The corresponding basis of $\R^{n+2}$ is denoted by $(E_0,\dots, E_{n+1})$ and is said the [*standard (light-cone) basis*]{} of $\R^{n+2}$. With respect to $(y_0,\dots , y_{n+1})$ the scalar product associated to $g$ can be written as $$\label{sp}\langle Y,Y'\rangle=-(y_0y'_{n+1}+y_{n+1}y'_0)-y_1y'_1+\sum_{j=2}^{n}y_jy'_j.$$ In addition, $dV=dy_0\wedge \dots \wedge y_{n+1}$ is a positive volume form. From now on we will use the light-cone coordinates and the elements $V\in \R^{n+2}$ are thought of as column vectors constructed from the components of $V$ with respect to $(E_0,\dots ,E_{n+1})$. For each $V\in \R^{n+2}$, $V\neq 0$, we denote by $[V]$ the oriented line spanned by $V$. The map $ V\in \mathcal{E}^{1,n-1}\to [V]$ allow us to identify $\mathcal{E}^{1,n-1}$ and the manifold of the isotropic oriented lines through the origin of $\R^{n+2}$ (null rays). Using such an identification, the connected component of the identity $\mathrm{A}^{\uparrow}_+(2,n)$ of the pseudo-orthogonal group of (\[sp\]) acts transitively and effectively on the left of $\mathcal{E}^{1,n-1}$ by $\mathbf{X}[V]=[\mathbf{X}\cdot V]$. The action preserves the oriented, time-oriented conformal Lorentzian structure of $\mathcal{E}^{1,n-1}$. It is 