Infinite Dimensional Path Bundles: Connections on the Infinite Dimensional Path Bundles of Finite Dimensional Principal Fiber Bundles

This thesis describes the construction of the so called path bundle. The starting point of the construction is a principal fiber bundle of finite dimensional manifolds. Next one considers the spaces of all mappings from a compact interval I into the base space and the total space of the bundle. These spaces are manifolds modelled over convenient vector spaces and are called manifolds of paths. Together with the Lie group of all paths in the structure group of the finite dimensional principal fiber bundle, one constructs a principal fiber bundle where the base and total spaces are infinite dimensional manifolds of paths. This is called the path bundle. After that one can consider natural subbundles of the path bundle. Next connections and connection forms on the path bundle are introduced and used to show the triviality of certain subbundles. Also it is discussed how connections on the finite dimensional principal fiber bundle induce connections on the path bundle and which properties they posses. Finally, curvature terms are defined. Furthermore as a special type of connections, the uniform ones are discussed together with a non trivial example.