  In this work we present a study on the characterization of ordered and
disordered hyperuniform point distributions on spherical surfaces. In spite of
the extensive literature on disordered hyperuniform systems in Euclidean
geometries, to date few works have dealt with the problem of hyperuniformity in
curved spaces. As a matter of fact, some systems that display disordered
hyperuniformity, like the space distribution of photoreceptors in avian retina,
actually occur on curved surfaces. Here we will focus on the local particle
number variance and its dependence on the size of the sampling window (which we
take to be a spherical cap) for regular and uniform point distributions, as
well as for equilibrium configurations of fluid particles interacting through
Lennard-Jones, dipole-dipole and charge-charge potentials. We will show how the
scaling of the local number variance enables the characterization of
hyperuniform point patterns also on spherical surfaces.
