Proper Almost-Homogeneous Domains of the Einstein Universe

The Einstein universe ${\mathbf{E}\mathbf{i}\mathbf{n}}^{p,q}$ of signature $(p,q)$ is a pseudo-Riemannian analogue of the conformal sphere; it is the conformal compactification of the pseudo-Riemannian Minkowski space. For $p,q\ge 1$, we show that, up to a conformal transformation, there is only one almost-homogeneous domain in ${\mathbf{E}\mathbf{i}\mathbf{n}}^{p,q}$ that is bounded in a suitable stereographic projection. This domain, which we call a diamond, is a model for the symmetric space of $\mathrm{PO}(p,1)\times \mathrm{PO}(1,q)$. We deduce a classification of closed conformally flat manifolds with proper development.