Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions

Abstract

We give an elementary projective geometry presentation of the classical Riemannian model spaces (elliptic and hyperbolic spaces) and of the classical Lorentzian model spaces (de Sitter and anti-de Sitter spaces). We also present some relevant degenerate model spaces (Euclidean and co-Euclidean spaces, Lorentzian Minkowski and co-Minkowski spaces), and geometric transitions. An emphasis is given to dimensions 2 and 3, convex subsets, duality, and geometric transitions between the spaces.