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

François Fillastre (1) , Andrea Seppi (2)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 IF - Institut Fourier

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.

Mots clés: Projective geometry; model spaces; convexity; duality; transition; degeneration; AMS code 51A05 52A15 52A55 53C45

Domaines: Géométrie différentielle [math.DG]; Topologie géométrique [math.GT]

François Fillastre, Andrea Seppi. Spherical, hyperbolic and other projective geometries: convexity, duality, transitions. Eighteen essays on non-Euclidean geometry (Vincent Albenge and Athanase Papadopoulos ed.). European Mathematical Society Publishing House., 2019. ⟨hal-01675487v2⟩