The Schrödinger structure of hyperbolical space-times recovered by embedded coordinates

We generalize the Schr\"odinger formalism which describes the embedding of the de Sitter spacetime in an euclidean spacetime with one more dimension. After presenting this formalism and its recovering to the Robertson-Walker metric, we present and discuss the embedding of a 5-sphere and a 5-hyperboloid.Introducing the static frame of the de Sitter metric, we write the reduced Schr\"odinger model in suitable coordinates. From the embedded projective coordinates, obtained by doing a stereographic projection of the Wick-rotated de Sitter 5-sphere, we consider the Beltrami (geodesic) representation, which gives a more general formulation of the seminal full model Schr\"odinger description of the geometry and topology of de Sitter spacetime. Our formalism retrieves the Schr\"odinger one if we consider the linear metric terms over the dS splitting on Minkowski spacetime.