Conformal compactification of spacetimes

The conformal groups for the nine two-dimensional real spaces of constant curvature are realized as matrix groups acting as globally defined linear transformations in a four-dimensional 'conforrnal ambient space'. This affords a unified and global study of the 'conformal completion' or compactification for the three classical Riemannian spaces as well as of the six relativistic and non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both Newton-Hooke and Galilean). The conformal embedding of the initial space into its compactification is carried out explicitly through two methods: either a group-theoretical one involving one-parameter subgroups or a geometric one by means of a stereographic projection. In the Euclidean and Minkowskian spaces the results reduce to the well known ones, but in the generic situation, with any non-zero curvature or arbitrary type signature, the approach is very explicit and provides some new insights.