  A recently-introduced function $\rho$ of spacetime event $P$ expressing
spacetime as made of 'spacetime atoms' of quantum origin is considered. Using
its defining relation, we provide an exact expression for $\rho$ involving the
van Vleck biscalar, and show it can be recast in terms of the extrinsic
curvature of suitable equi-geodesic surfaces centered at $P$. Moreover, looking
at the role $\rho$ plays in the statistical description of spacetime, we point
out that this quantity should actually be understood as counting the quantum
states of the collection of spacetime atoms rather than counting directly the
spacetime atoms themselves (or the degrees of freedom associated to them), and
would correspond to the ratio of the number of quantum states 'at $P$' for an
assigned spacetime configuration to the number of quantum states for flat
spacetime.
