Title:

Some aspects of curvature in general relativity

The purpose of this thesis is to study in depth the relationship between the curvature of spacetime and the other geometrical objects which naturally arise in general relativity. Most of the results obtained apply to the generic case. Chapter 1 contains a discussion of certain aspects of fibre bundle theory required in later chapters which may be unfamiliar to many relativists, while chapter 2 contains preliminary material on curvature in relativity and proves a continuity property of the algebraic classification of the Weyl and energymomentum tensors. Chapter 3 describes the generic behaviour of the Riemann, Weyl and energymomentum tensors, and chapter 5 goes on to use this description to investigate the relationship of the Riemann tensor to the metric, conformal class and connection of spacetime in the generic case. In particular it is proved that the Riemann tensor uniquely and continuously determines the connections. The information obtained in chapter 3 on the algebraic type of curvature in the general case is related in chapter 4 to the topology of the underlying manifold. In chapter 6 a topology is defined on the set of sectional curvatures of all Lorentz metrics on a given manifold. The remainder of the chapter attempts to do for the sectional curvature what was done for the Riemann tensor in chapter 5 but, because sectional curvature is more difficult to handle, the results obtained are necessarily more modest.
