  This work is intended to investigate the geometry of anti-de Sitter spacetime
(AdS), from the point of view of the Laplacian Comparison Theorem (LCT), and to
give another description of the hyperbolical embedding standard formalism of
the de Sitter and anti-de Sitter spacetimes in a pseudo-Euclidean spacetime. It
is shown how to reproduce some geometrical properties of AdS, from the LCT in
AdS, choosing suitable functions that satisfy basic properties of Riemannian
geometry. We also introduce and discuss the well-known embedding of a 4-sphere
and a 4-hyperboloid in a 5-dimensional pseudo-Euclidean spacetime, reviewing
the usual formalism of spherical embedding and the way how it can retrieve the
Robertson-Walker metric. With the choice of the de Sitter metric static frame,
we write the so-called reduced model in suitable coordinates. We assume the
existence of projective coordinates, since de Sitter spacetime is orientable.
  From these coordinates, obtained when stereographic projection of the de
Sitter 4-hemisphere is done, we consider the Beltrami geodesic representation,
which gives a more general formulation of the seminal full model described by
Schrodinger, concerning the geometry and the topology of de Sitter spacetime.
Our formalism retrieves the classical one if we consider the metric terms over
the de Sitter splitting on Minkowski spacetime. From the covariant derivatives
we find the acceleration of moving particles, Killing vectors and the isometry
group generators associated to de the Sitter spacetime.
