In this talk, we introduce, briefly, the basis of Lorentzian geometry and cohomogeneity one actions. We also, introduce the Einstein universe and consider an interesting cohomogeneity one action on $\mathbb{Ein}^{1,2}$.
A smooth manifold $M$ equipped with a metric tensor $\textbf{g}$ of signature $(1, n)$ is called a Lorentzian manifold. Precisely, the restriction of $\textbf{g}$ to the tangent space $T_pM$ is of the form $−x_1^2+x_2^2+\cdots+x_n^2$.
Let $G$ be a Lie group which acts on a smooth manifold. Then the orbit of $G$ at point $p\in M$ is $G(p) := \{gp : g \in G\}$. If $G$ acts on $M$ smoothly, then every orbit of $G$ is a smooth immersed submanifold of $M$. The action of $G$ on $M$ is called cohomogeneity one if $G$ admits a codimension 1 orbit in $M$. Furthermore, if $M$ is a pseudo-Riemannian manifold and $G$ acts on $M$ isometrically, then every orbit is a pseudo-Riemannian submanifold. In other words, the restriction of the metric on each orbit has constant signature.
Consider the direct product Lorentzian manifold $(M, \textbf{g}) = (\mathbb{S}^1\times \mathbb{S}^n, −d\theta^2+\textbf{g}_{\mathbb{S}^n}$ ) where $(\mathbb{S}^1, d\theta^2)$ is the usual Riemannian circle of radius 1 and $(\mathbb{S}^n, \textbf{g}_{\mathbb{S}^n})$ is the usual Riemannian $n$-dimensional sphere of constant sectional curvature 1. The map $-Id : M \to M$ sending $(x, y)$ to $(-x, -y)$ is an isometry. The $(n + 1)$-dimensional Einstein universe $\mathbb{Ein}^{1,n}$ can be defined as the quotient of $M$ by $\{Id, -Id\}$. The Einstein universe is Lorentzian analogue of the sphere $\mathbb{S}^n$. It compactifies the Minkowski space and is the conformal boundary of Anti de Sitter space.