Projective symmetries, holonomy and curvature structure in general relativity
In the mathematical study of general relativity it is valuable to consider the classification and properties of the metric connection and Riemannian curvature derived from the spacetime metric. The aim of this thesis is to present various results obtained regarding (i) the holonomy groups of simply-connected spacetimes (ii) the problem of deciding if a connection is a metric connection (iii) a sub-class of curvature preserving symmetries and (iv) geodesic preserving symmetries. The holonomy group associated with a spacetime provides a global classification scheme for the metric connection. After the necessary background and definitions, chapter three provides an analysis of the holonomy groups possible in many of the commonly studied classes of spacetime. While every metric is uniquely associated with its metric connection, the converse is not true in general. The question of when a symmetric connection and the curvature derived from it can be associated with a Lorentz signature metric can be addressed via the existence of integrability conditions involving a metric candidate. In chapter four such an analysis is performed in terms of a classification scheme based on the rank and eigenbivector structure of the curvature. The problem is resolved in that the maximum number of such integrability conditions required to guarantee that the connection is a metric connection is given for each (non-trivial) curvature type. Transformations of a spacetime that preserve some geometric quantity or structure are particularly important in general relativity. In chapter five some results regarding transformations that preserve the curvature and its covariant derivatives are presented for spacetimes in which the preservation of the curvature alone is not enough to completely specify the nature of the curvature symmetry.