Polygons of the Lorentzian plane and spherical simplexes

François Fillastre (1)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed and endowed with a suitable area, is isometric to a spherical polyhedron. These polygons have an infinite number of vertices, are space-like, contained in the future cone of the origin, and setwise invariant under the action of a linear isometry.

Domaines: Géométrie différentielle [math.DG]

François Fillastre. Polygons of the Lorentzian plane and spherical simplexes. Elemente der Mathematik, 2014, 69 (3), pp.144--155. ⟨10.4171/EM/256⟩. ⟨hal-00640924v2⟩