Geometrization of the gauge connection within a Kaluza-Klein theory

We geometrize a generic (abelian and non-abelian) gauge coupling within the framework of a Kaluza-Klein theory, by choosing a suitable matter- field dependence on the extra coordinates. We first extend the Nother theorem to a multidimensional spacetime, the Cartesian product of a 4-dimensional Minkowski space and a compact homogeneous manifold (whose isometries reflect the gauge symmetry). On such a "vacuum" configuration, the extra-dimensional components of the field momentum correspond to the gauge charges. Then we analyze the structure of a Dirac algebra for a spacetime with the Kaluza-Klein restrictions. By splitting the corresponding free-field Lagrangian, we show how the gauge coupling terms arise.