The Funk and Hilbert geometries for spaces of constant curvature

Abstract

The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets $\Omega$ of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representations of it in each of the non-Euclidean cases, which parallel the known situation for the Euclidean case. As a consequence, all of these metrics are shown to be Finslerian, and the associated norms of the Finsler metrics are described. The theory is developed by using a set of classical trigonometric identities on the sphere $S^n$ and the hyperbolic space $\mathbb{H}^n$ and the definition of a cross ratio on the non-Euclidean spaces of constant curvature. This in turn leads to the concept of projectivity invariance in these spaces. We then study the geodesics of the Funk and Hilbert metrics. In the case of Euclidean (respectively spherical, hyperbolic) geometry, the Euclidean (respectively spherical, hyperbolic) geodesics are Funk and Hilbert geodesics. Natural projection maps that exist between the spaces $\mathbb{R}^n$, $\mathbb{H}^n$ and the upper hemisphere demonstrate that the theories of Hilbert geometry of convex sets in the three spaces of constant curvature are all equivalent. The same cannot be said about the Funk geometries.