A study of certain linear connections arising in physical theories with particular reference to holonomy
The aim of this thesis is to study certain linear connections arising in physics; and in particular metric connections, Weyl connections and Cartan connections are examined. Emphasis is placed on the holonomy group of the connection and the relationships between the relevant geometric objects in the physical theories. The appropriate mathematical background is reviewed in the first two chapters and various notions from differential geometry are introduced. Proofs of theorems relating covariantly constant and recurrent tensors with holonomy are given in detail, and Eisenhart's results on connections induced in submanifolds are given a modern treatment. The relationship between a metric, its Levi-Civita connection and its curvature tensor is examined in chapter three. Some new results on the problem of how to 'recognise' a metric curvature are presented and the idea of a 'curvature copy' - different linear connections with the same curvature - is discussed. Weyl's elegant attempt to unify gravity with electromagnetism is the topic of the fourth chapter. A full holonomy classification of Weyl connections is given in this chapter, along with results concerning the relationship between metric, connection and curvature. The Cartan connection was introduced by Cartan as a device for placing Newtonian gravity in a similar formal setting to Einstein's theory of gravity. This connection is studied in chapter five and reference is made to holonomy properties, symmetries and reduction of the bundle - which enables Newtonian gravity to take on the appearance of a gauge field theory.