1991 MSC: 17B70, 58E99, 55R10

Keywords: Froelicher-Nijenhuis bracket, Principal bundles

We study the basic aspects of the theory of systems of tangent-valued forms for the particular case of principal bundles, using the Froelicher-Nijenhuis bracket.

In the recent years, a general theory of tangent-valued forms, based on the Froelicher-Nijenhuis bracket, has been developed. This approach enables us to define and study the most fundamental features independently from any symmetry group acting on the fibres, and gives rise naturally to universal structures. An important clarification is then achieved on the precise roles of the various assumptions: the fibre structure (e.g. principal bundle structure) selects a distinguished system of tangent-valued forms. There are important subalgebras, whose relation to the system of principal connections will be examined in a forthcoming paper. The case of principal bundles, fundamental in many applications, is in principle a particular case of the general theory of systems. It is thus interesting to work out explicitely this relation, because we achieve a clarification of the geometry of principal bundles and, at the same time, give an expressive example of the general theory. However, such a job would be not immediate for the average reader, who may also may find some difficulty in tackling directly the papers on the general theory. Thus we purposed to give a new introduction to this subject, keeping the philosophy and methodology of the general treatment but developping it explicitely for principal bundles; when appropriate we stress the equivalence or the differences with the traditional expositions.