Aspects of Quasi-Riemannian Geometries

The aim of this thesis is to examine to some extent the physical implications of quasi-Riemannian theories, with particular attention to the modifications they introduce in higher dimensional theories of gravitation and to the applications to the Ka l uza-Kl ei n theories. With the matter of investigation so vast, we preferred to study various features of the theory rather than specialize on one single subject. How ever, we limited ourselves to the case of a tangent space group of the kind 0( 1,N-1 )xOCM). This is the simplest nontrivial choice but nevertheless it contains a 11 the essential features of the quasi-R i em anni an geometries. The plan of the thesis is as follows. In chapter I a short review of differential geometry and general relativity in higher dimensions is given, with special attention to the problems related to the choice of the tangent space group. The special features of the Riemannian geometry compared with more general structures are emphasized. Chapter 11 contains a brief review of standard Kaluza-Klein theories. The problem of the fermion chirality is examined and some simple examples of spontaneous corn pact if i cation are discussed. In the next chapters quas i-R i em ann i an theories are studied in great detail. Most of the material contained there is based on my contributions to the subject either a 1 ready published or being prepared for publication. In particular, in chapter 111 the most general action for quasiRiemannian theories with tangent space 0( 1,N-1 )xO(M) compatible with some simple phys i ea l requ trem ents is established. It is shown to depend on 9 independent parameters, and is compared with the actions obtained from different approaches to the theory. In chapter IV the stability of the flat space under sm a 11 perturbations and the particle content of the theory are studied and it is shown that some very strong conditions must be imposed on the parameters of the theory in order to achieve stability. In fact, a gauge invariance must be introduced at the linear level, in order to avoid the appearance of ghost states. In chapter V, the investigation is briefly extended to the cl ass i ea l fields defined on a quasi-Riemannian background and the definition of the metric and of the geodesics is discussed. Chapter VI is devoted to the study of the solutions of the classical field equations stemming from the action introduced in chapter Ill. In particular, a quasi-Riemannian cosmological model is described and compared with its Riemannian limit. Also the possible generalizations of the Schwarzschild solution of general relativity are discussed. Finally, chapter VI I deals w.ith the applications of quasi-Riemannian geometries to Kaluza-Klein theories. Some simple models exhibiting spontaneous corn pact if i cation are introduced and the "zero-mode ansatz" is discussed. Unfortunately, the results are not satisfactory from a phenomenological point of view. A discussion on the possibility of obtaining more realistic models by using different tangent space groups concludes the chapter.