Affine and curvature collineations in space-time
The purpose of this thesis is the study of the Lie algebras of affine vector fields and curvature collineations of space-time, the aim being, in the first case, to obtain upper bounds on the dimension of the Lie algebra of affine vector fields (under the assumption that the space-time is non-flat) as well as to obtain a characterization of such vector fields in terms of other types of symmetries. In the case of curvature collineations the aim was that of characterizing space-times which may admit an infinite-dimensional Lie algebra of curvature collineations as well as to find local characterizations of such vector fields. Chapters 1 and 2 consist of introductory material, in Differential Geometry (Ch.l) and General Relativity (Ch.2). In Chapter 3 we study homothetic vector fields which admit fixed points. The general results of Alekseevsky (a) and Hall (b) are presented, some being deduced by different methods. Some further details and results are also given. Chapter 4 is concerned with space-times that can admit proper affine vector fields. Using the holonomy classification obtained by Hall (c) it is shown that there are essentially two classes to consider. These classes are analysed in detail and upper bounds on the dimension of the Lie algebra of affine vector fields of such space-times are obtained. In both cases local characterizations of affine vector fields are obtained. Chapter 5 is concerned with space-times which may admit proper curvature collineations. Using the results of Halford and McIntosh (d) , Hall and McIntosh (e) and Hall (f) we were able to divide our study into several classes The last two of these classes are formed by those space-times which admit a (1 or 2-dimensional) non-null distribution spanned by vector fields which contract the Riemann tensor to zero. A complete analysis of each class is made and some general results concerning the infinite-dimensionality problem are proved. The chapter ends with some comments in the cases when the distribution mentioned above is null.