Some Remarks on the AdS Geometry, Projective Embedded Coordinates and Associated Isometry Groups

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 pseudoeuclidian spacetime. After Witten proved that general relativity is a renormalizable quantum system in (1+2) dimensions, it is possible to point out a few interesting motivations to investigate the AdS spacetime. A lot of attempts were made to generalize the gauge theory of gravity in (1+2) dimensions to higher ones. The first one was to enlarge the Poincar\'e group of symmetries, supposing an AdS group symmetry, which contains the Poincar\'e group. Also, the AdS/CFT correspondence asserts that a maximal supersymmetric Yang-Mills theory in 4-dimensional Minkowski spacetime is equivalent to a type IIB closed superstring theory. The 10-dimensional arena for the type IIB superstring theory is described by the product manifold $S^5\times$ AdS, an impressive consequence that motivates the investigations about the AdS spacetime in this paper, together with the de Sitter spacetime. Classical results in this mathematical formulation are reviewed in a more general setting together with the isometry group associated to the de Sitter spacetime. It is known that, out of the Friedmann models that describe our universe, the Minkowski, de Sitter and anti-de Sitterspacetimes are the unique maximally isotropic ones, so they admit a maximal number of conservation laws and also a maximal number of Killing vectors. In this paper 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 pseudoeuclidian 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 Schr\"odinger, concerning the geometry and topology of de Sitter spacetime. Our formalism retrieves the classical one if we consider the linear 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.