Conformal compactification and cycle-preserving symmetries of spacetimes

The cycle-preserving symmetries for the nine two-dimensional real spaces of constant curvature are collectively obtained within a Cayley–Klein framework. This approach affords a unified and global study of the conformal structure of 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), and gives rise to general expressions holding simultaneously for all of them. Their metric structure and cycles (lines with constant geodesic curvature that include geodesics and circles) are explicitly characterized. The corresponding cyclic (Möbius-like) Lie groups together with the differential realizations of their algebras are then deduced; this derivation is new and much simpler than the usual ones and applies to any homogeneous space in the Cayley–Klein family, whether flat or curved and with any signature. Laplace and wave-type differential equations with conformal algebra symmetry are constructed. Furthermore, the conformal groups are realized as matrix groups acting as globally defined linear transformations in a four-dimensional ‘conformal ambient space’, which in turn leads to an explicit description of the ‘conformal completion’ or compactification of the nine spaces.