A Geometric Approach to Noncommutative Principal Bundles

The question of whether there is a way to translate the geometric concept of fibre bundles to Noncommutative Geometry is quite natural and highly interesting. In the case of vector bundles the Theorem of Serre and Swan gives the essential clue. The case of principal bundles is not treated in a satisfying way. From a geometrical point of view it is, so far, not sufficiently well understood what should be a ``noncommutative principal bundle''. Still, there is a well-developed abstract algebraic approach using the theory of Hopf algebras. An important handicap of this approach is the ignorance of any topological and geometrical aspects. The aim of this thesis is to develop a geometrically oriented approach to the noncommutative geometry of principal bundles based on dynamical systems and the representation theory of the corresponding transformation groups.