Gauge-reversing maps on cones, and Hilbert and Thompson isometries

Abstract

We show that a cone admits a gauge-reversing map if and only if it is a symmetric cone. We use this to prove that every isometry of a Hilbert geometry is a collineation unless the Hilbert geometry is the projective space of a non-Lorentzian symmetric cone, in which case the collineation group is of index two in the isometry group. We also determine the isometry group of the Thompson geometry on a cone.

36 pages, 3 figures

References in corpus (1)

• The horofunction boundary of the Hilbert geometry

Cited by in corpus (3)

• Horofunction compactifications and duality
• A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains
• Horofunctions and metric compactification of noncompact Hermitian symmetric spaces