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

Cormac Walsh (1)

1 TROPICAL - TROPICAL

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.

Domaines: Géométrie métrique [math.MG]

Cormac Walsh. Gauge-reversing maps on cones, and Hilbert and Thompson isometries. Geometry and Topology, 2018, 22 (1), pp.55-104. ⟨10.2140/gt.2018.22.55⟩. ⟨hal-00930929⟩