  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.
