Space time of class one

The authors study space-times embedded in E{sub 5} (that means, pseudo-euclidean five-dimensional spaces) in the intrinsic rigidity case, i.e., when the second fundamental form b{sub if} can be determined by the internal geometry of the four-dimensional Riemannian space R{sub 4}. They write down the Gauss and Codazzi equations determining the local isometric embedding of R{sub 4} in E{sub 5} and give some consequences of it. They prove that when there exists intrinsic rigidity, then b{sub if} is a linear combination of the metric and Ricci tensor; it is given some applications for the de Sitter and Einstein models.