About the generalisation of the Kaluza-Klein theory

The goal of this paper is to present the Kaluza-Klein theory. In the first part we will discuss the theory elaborated by Kaluza and Klein, in a Riemannian space with five dimensions, which unifies the gravitation with electromagnetism. The second part debates the generalization of this theory in a space with 4+n dimensions. This is a mathematical product between the Riemannian 4-dimension variety and the G/H n-dimensional homogenous space. In the last part we will propose a Kaluza-Klein like theory in the fiber bundle space with 4+n dimensions. Every part is structured as follows: we will identify the metric tensor G for the gravitation and the potentials Yang-Mills; then we will deduce the equations of geodesics and the equations of the field. (author)