Crofton formulae and geodesic distance in hyperbolic spaces

TL;DR

This paper explores integral formulae for geodesic distances in hyperbolic spaces, demonstrating their properties as kernels of negative type and contrasting with projective spaces.

Contribution

It introduces new integral formulae for geodesic distances in real and complex hyperbolic spaces, highlighting their negative type properties.

Findings

Geodesic distance in real hyperbolic space is a hypermetric and negative type kernel.

An integral formula involving hyperplanes is used for real hyperbolic space.

An analogous formula involving horospheres is provided for complex hyperbolic space.

Abstract

The geodesic distance between points in real hyperbolic space is a hypermetric, and hence is a kernel negative type. The proof given here uses an integral formula for geodesic distance, in terms of a measure on the space of hyperplanes. An analogous integral formula, involving the space of horospheres, is given for complex hyperbolic space.By contrast geodesic distance in a projective space is not of negative type.