1991 MSC: 53C05, 55R10, 58A20

Keywords: Froelicher-Nijenhuis bracket, Principal bundles, Connections

We study the basic aspects of the theory of systems of connections and overconnections for the case of principal bundles, using the Froelicher-Nijenhuis algebra of tangent-valued forms, which was examined in this context in [a13]. We discuss the relation between the horizontal and vertical approaches and the traditional approach based on Lie algebra-valued forms.

There exists a rich literature on principal bundles and invariant connections, stimulated by their importance in geometry and physics. Many important results have been achieved in the traditional approach based on the Maurer-Cartan formulas, in which the group action plays an essential role at any stage and at any level. However, in the recent years, a more general approach to the theory of connections, based on the Froelicher-Nijenhuis bracket of tangent-valued forms, has been developed. In a previous paper [a13] we examined the theory of tangent-valued forms for the particular case of principal bundles, stressing how the general approach yields an important clarification of the picture; here we shall apply it to connections, overconnections and related topics. Furthermore, we make a detailed comparison with the traditional approach. All notations and preliminary results have been introduced in [a13].