GENERAL REMARKS AND PREREQUISITES 1.1 Introduction 1.2 Differentiable manifolds 1.3 Riemannian manifolds Metrics, connections and curvatures Particular spaces |
1-1 Introduction Physical motivations
It is nowadays believed that it is possible and useful to describe physics in an "extended" space-time. Events that we see and that we measure are usually described by 3+1 numbers labeling the position and the time. Forces acting on objects and influencing their trajectories have also been described in the past in term of various tensor fields defined on a four dimensional manifold modeling the "space of events". It happens that, in many cases, the theory takes a simpler form if we assume that what we observe is just a shadow (projection) of something that takes place in space-time which has more than 4 dimensions. As an example, it has been recognized long ago that coupled gravitational and electromagnetic fields respectively described by a (4 dimensional) hyperbolic metric and a Maxwell field (a U( 1 ) connection), could be also described by a U(1 ) invariant metric on a five-dimensional space. It is very possible (and it is the belief of the authors) that a correct formalization of the physics of "our universe" should involve an infinite dimensional manifold, and that for reasons which are still unknown, what we see classically looks four-dimensional. The fact that we do not see the extra dimensions (those of the so called "internal space" ) can be described, if not explained, by the fact that the metric of our multidimensional universe singles out some directions along which it is invariant or at least equivariant (in some sense). Many papers have been published recently in the physical literature presenting many different constructions, sometimes under the same headings (Kaluza-Klein theories, Dimensional reduction, Symmetries of gravitational and gauge fields , etc.); very often the generality of the described situation was not studied. One of the aims of the present book is to present the geometrical and analytical aspects of "dimensional reduction" and to discuss with more generality several situations which have been considered in the past. Content of the book What the book is really about is Riemannian geometry of those spaces on which a group action is given (with a view on applications to physical theories -"unified theories"-). This study involves in particular the geometry of group manifolds, homogeneous spaces, principal bundles, non principal bundles (with group action),.., but also, in order to study the different kinds of "fields" defined on those spaces, it requires an appropriate generalization of (non abelian) harmonic analysis. Each chapter of the book begins with a summary section which stresses the main ideas in plain terms the reader willing to make his knowledge more precise should then read the rest of the chapter where a more detailed discussion (using a more precise mathematical language) is given. The summary introductions do not usually require any knowledge of fiber bundles; however, we use freely the corresponding terminology and results in the core of each chapter. Indeed, although the summary section usually describes everything in a "local" way (e.g. using coordinates ), we always want to render our considerations global. The remaining sections of this first chapter recall some standard definitions of differential (Riemannian) geometry and has also the purpose of setting our conventions. Most of the results discussed here will be used freely in all chapters of the book; however, we should mention that the reader who wants to recast Riemannian geometry in the general framework of the theory of connections should jump directly to Ch.6, where these notions are developed from scratch. Anybody willing to construct physical models generalizing the "old" Kaluza-Klein ideas should be first acquainted with some basic facts about the Riemannian structure(s) of Lie groups (Ch.2) and homogeneous spaces (Ch.3). The study of G-invariant metrics on groups and homogeneous spaces is also compulsory if one wants to analyze the situation when the space is a (generally only local) product of some manifold M times a group G or a homogeneous space G/H. G-invariant metrics on principal bundles and non-principal bundles carrying a G action are discussed respectively in Ch.4 and Ch.5. A general study of the Riemannian geometry of "matter fields", i.e., vector valued functions (or forms) defined on a manifold (in particular the covariant derivative acting on tensors, spinors, p-forms valued in some vector space...) is made in Ch.6 (this chapter could be read independently of the rest of the book). It is well known that, when a real (or complex) valued function is defined on a group or on a homogeneous space, it is possible to "expand" it (think of the usual spherical harmonics); however, when the underlying space is only a (local) product of some manifold M by G or G/H, the formalism has to be generalized and this is done in Ch.7. The particular case where such matter fields are usual tensors or spinors is studied in Ch.8 (G-spin structures are naturally obtained there as a result of a process of "dimensional reduction"). The techniques described in particular in chapters 5, 7 and 8 provide us with a "general-purpose-tool" that we may use in several situations; as an example of such a use, we study in Ch.9 the dimensional reduction of Einstein-Yang-Mills systems i.e. analyze the geometry of a manifold on which both metric and connection are given, along with the action of a symmetry group. We will study this case by showing how it can be reduced to the situation studied in Ch.5. Finally, in Ch.10, we consider a more general situation where there is no global action of a finite dimensional Lie group G but where we can nevertheless define "interesting metrics" which are invariant under a "bundle of groups" (infinite dimensional groups of automorphisms of bundles are defined and studied in section 4.1 1). Each main section of the book ends with a paragraph entitled "Pointers to the literature"; indeed, references are usually not given within each chapter but collected at the end. New results Before ending this introduction, we should maybe mention what is "original" in this book and what can be found elsewhere. It is clear that the whole discussion of homogeneous metrics on Lie groups and coset spaces can be found in a scattered way inside many papers of the mathematical literature. However the general discussion of metrics leading to "dimensional reduction", and in particular the general study of metrics on bundles with homogeneous fibers is probably new: although known "in principle", many explicit constructions and calculations carried out here do not seem to have been discussed elsewhere in the mathematical or in the physical literature (but by the authors themselves). Also, let us mention a few other mathematical (or physical) constructions hardly to be found elsewhere: generalization of Frobenius and Peter-Weyl theorems (in Ch.7), intrinsic definition of the Lichnerowicz operator (in Ch.6), nonstandard discussion of Einstein-Cartan theory with spinors (in Ch.6), link between G-spin structures and dimensional reduction (in Ch.8), generalization of the Wang theorem on G-invariant connections (in Ch.9), definition and study of "local" action of groups (bundle of groups, in Ch.4.11 and Ch. 10 ). |
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