Holonomy and the determination of metric from curvature in general relativity
In a large class of space-times, the specification of the curvature tensor components Rabcd in some coordinate domain of the space-time uniquely determines the metric up to a constant conformal factor. The purpose of this thesis is to investigate the spaces where the metric is not so determined, and to look at the determination of the metric when the components of the derivatives of the Riemann tensor (one index up) are also specified, with special reference to the role of the infinitesimal holonomy group (ihg). In chapter one we set up the mathematical background, describing the Weyl and Ricci tensor classifications, and defining holonomy. In chapter two we look at spaces with Riemann tensors of low rank. This leads us on to decomposable spaces and the connection between decomposable spaces and relativity in three dimensions. We examine the connection between decomposability and the ihg, and relate this to the Weyl and Ricci tensor classifications. In chapter three we discuss the problem of determination of the metric by the Riemann tensor alone, and give a brief review of the history of the problem. In chapter four we go on to look at the determination of the metric by the curvature and its derivatives. It is shown that, with the exception of the generalised pp-waves, we only need look as far as the first derivatives of the Riemann tensor to obtain the best determination of the metric, unless the Riemann tensor is rank 1, when the second derivatives may also be required. The form of the metric ambiguity, the ihg and Petrov types are determined in each case. These results are then reviewed in the final chapter.