Funk and Hilbert geometries in spaces of constant curvature

Abstract

We survey the Funk and Hilbert geometries of open convex sets in the sphere $S^n$ and in the hyperbolic space ${\mathbb{H}}^n$. The theories are developed in analogy with the classical theory in Euclidean space. It is rather unexpected that the Funk geometry, whose definition and development use the affine structure of Euclidean space, has analogues in the non-linear spaces $S^n$ and ${\mathbb{H}}^n$, where there are no analogies. As we shall see, the existence of a Funk geometry in these non-linear spaces is based on some non-Euclidean trigonometric formulae which display some kind of similarity between (the hyperbolic sine of the lengths of) sides of right triangles. The Hilbert metric in each of the constant curvature settings is a symmetrization of the Funk metric. We show that the Hilbert metric of a convex subset in a space of constant curvature can also be defined using a notion of a cross ratio which is proper to that space.