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Motivated by various higher dimensional theories in high-energy-physics and cosmology, we consider the local and global isometric embeddings of pseudo-Riemannian manifolds into manifolds of higher dimensions. We provide the necessary background in general relativity, topology and differential geometry, and present the technique for local isometric embeddings. Since an understanding of the local results is key to the development of global embeddings, we review some local existence theorems for general pseudo-Riemannian embedding spaces. In order to gain insight we recapitulate the formalism required to embed static spherically symmetric space-times into fivedimensional Einstein spaces, and explicitly treat some special cases, obtaining local and isometric embeddings for the Reissner-Nordstr¨om space-time, as well as the null geometry of the global monopole metric. We also comment on existence theorems for Euclidean embedding spaces. In a recent result, it is claimed (Katzourakis 2005a) that any analytic n-dimensional space M may be globally embedded into an Einstein space M × F (F an analytic real-valued one-dimensional field). As a corollary, it is claimed that all product spaces are Einsteinian. We demonstrate that this construction for the embedding space is in fact limited to particular types of embedded spaces. We analyze this particular construction for global embeddings into Einstein spaces, uncovering a crucial misunderstanding with regard to the form of the local embedding. We elucidate the impact of this misapprehension on the subsequent proof, and amend the given construction so that it applies to all embedded spaces as well as to embedding spaces of arbitrary curvature. This study is presented as new theorems. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2007.

The holonomy group arising from a linear connection and differential homotopy is a classical subject in geometry. The notion was generalized first by Y. Muto ({10}) by considering horizontal subspaces in a fibred space which by construction is a differential manifold over a base space with another manifold as the fibre. He called this generalized group the restricted holonomy group Hl('o)((')M). Unlike the case of frame bundles the horizontal subspaces in a fibred space do not in general obey the right invariant rule. Hence it is not hard to imagine that Hl('o)((')M) is larger than linear holonomy groups. It may not even form a Lie group and for years the structure of this group was left unknown simply because the number of elements concerned is too large to handle. / One of the intentions here is to clarify and determine the structure of Hl('o)((')M) by setting certain conditions. Then by use of Palais' theorem about transformation groups, Nijenhuis' method for dealing with linear holonomy groups, and the standard technique of computing line integrals, the structure of Hl('o)((')M) is determined in Chapter One under certain conditions. Some properties concerning the isometric immersion from one fibred Riemannian space into another are also discussed in Chapter Two. / As far as I know, the work in this thesis is original, except where the text indicates the contrary: In particular, Chapter One is purely expository.

Bibliography: leaves 103-105. / xii, 105 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1997

Thesis (doctoral)--Universität Bonn, 1993. / Includes bibliographical references (p. 80-82).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 41-42).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2006 / Includes bibliographical references (p. 59-60).

Thesis (Dr. rer. nat.)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2001. / Includes bibliographical references (p. 87-90).

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Sergei Prokofiev is among the ranks of early-twentieth century composers whose music endures in the concert hall and whose life has attracted much musicological research. Fewer studies, however, have undertaken an analytical investigation into his music, and the body of scholarly work on the musical theoretical issues raised by his compositions does not rival that exploring the music of such major contemporaries as Igor Stravinsky and Béla Bartók. Existing Anglo-American contributions to the field of Prokofiev theory have mostly employed the tools of common-practice musical analysis, many of them using Schenkerian methods, with the more recent accounts of Richard Bass and Deborah Rifkin expanding these to incorporate the chromatic features of Prokofiev’s music in more sophisticated ways. A notable exception is Neil Minturn, who proposes an analytical approach informed by pitch-class set theory; his methodology, however, has not been developed in any further research. This thesis aims to make a contribution to Prokofiev analysis by applying recent developments in neo-Riemannian theories and the work of the noted Russian musicologist, Yuri Kholopov, whose early monograph on Prokofiev’s harmony has not been engaged with in English language accounts to date. Neo-Riemannian theories are well suited to this task due to the correspondence between their remit and the diatonic chromatic aspect of Prokofiev’s music. This thesis also introduces and explores the potential of Kholopov’s theoretical concepts regarding the nature of twentieth-century music, and in particular processes such as polyharmony, in original analytical applications. Prokofiev’s Symphony No. 5 and Piano Sonata No. 8 have been selected as focal works as they are acknowledged masterworks on an ambitious scale and arguably represent a shift in Prokofiev’s compositional thinking towards more abstract music in his later period. Existing analyses of extracts from these two works also offer the opportunity of making comparative observations. By focusing on harmony and large scale tonal design in these two works, this thesis hopes to demonstrate that a dialogue between the theoretical perspectives of Kholopov and those of neo-Riemannian theories may contribute valuable insights into Prokofiev’s music, at both surface and deep structural levels.