Lorentzian similarity manifold

If an $m+2$-manifold $M$ is locally modeled on ${\text{RR}}^{m+2}$ with coordinate changes lying in the subgroup $G={\text{RR}}^{m+2}⋊(\text{rO}(m+1,1)\times {\text{RR}}^{+})$ of the affine group $\text{rA}(m+2)$, then $M$ is said to be a \emph{Lorentzian similarity manifold}. A Lorentzian similarity manifold is also a conformally flat Lorentzian manifold because $G$ is isomorphic to the stabilizer of the Lorentz group $\text{rPO}(m+2,2)$ which is the full Lorentzian group of the Lorentz model $S^{2n+1,1}$. It contains a class of Lorentzian flat space forms. We shall discuss the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations.