The geometrical setting of gauge theories of the Yang-Mills type

The appropriate language for describing the pure Yang-Mills theories is introduced. An elementary but precise presentation of the mathematical tools which are necessary for a geometrical description of gauge fields is given. After recalling basic notions of differential geometry, it is shown in what sense a gauge potential is a connection in some fiber bundle, and the corresponding gauge field the associated curvature. It is also shown how the global aspects of the theory (e.g., boundary conditions) are coded into the structure of the bundle. Gauge transformations and equations of motion, as well as the self-duality equations, acquire then a global character, once they are defined in terms of operations in the bundle space. Finally the orbit space, that is to say, the set of gauge inequivalent potentials, is defined, and its is shown why there is no continuous gauge fixing in the non-Abelian case.