Global embeddings of pseudo-Riemannian spaces.

Abstract

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.