  This paper introduces a new modeling framework for the statistical analysis
of point patterns on a manifold M_{d}, defined by a connected and compact
two-point homogeneous space, including the special case of the sphere. The
presented approach is based on temporal Cox processes driven by a
L^{2}(\mathbb{M}_{d})-valued log-intensity. Different aggregation schemes on
the manifold of the spatiotemporal point-referenced data are implemented in
terms of the time-varying discrete Jacobi polynomial transform of the log-risk
process. The n-dimensional microscale point pattern evolution in time at
different manifold spatial scales is then characterized from such a transform.
The simulation study undertaken illustrates the construction of spherical point
process models displaying aggregation at low Legendre polynomial transform
frequencies (large scale), while regularity is observed at high frequencies
(small scale). K-function analysis supports these results under temporal
short-, intermediate- and long-range dependence of the log-risk process.
