Symmetries of Spacetimes and Riemannian Manifolds (Mathematics and Its Applications, 487)

This book provides an upto date information on metric, connection and curva­ ture symmetries used in geometry and physics. More specifically, we present the characterizations and classifications of Riemannian and Lorentzian manifolds (in particular, the spacetimes of general relativity) admitting metric (i.e., Killing, ho­ mothetic and conformal), connection (i.e., affine conformal and projective) and curvature symmetries. Our approach, in this book, has the following outstanding features: (a) It is the first-ever attempt of a comprehensive collection of the works of a very large number of researchers on all the above mentioned symmetries. (b) We have aimed at bringing together the researchers interested in differential geometry and the mathematical physics of general relativity by giving an invariant as well as the index form of the main formulas and results. (c) Attempt has been made to support several main mathematical results by citing physical example(s) as applied to general relativity. (d) Overall the presentation is self contained, fairly accessible and in some special cases supported by an extensive list of cited references. (e) The material covered should stimulate future research on symmetries. Chapters 1 and 2 contain most of the prerequisites for reading the rest of the book. We present the language of semi-Euclidean spaces, manifolds, their tensor calculus; geometry of null curves, non-degenerate and degenerate (light like) hypersurfaces. All this is described in invariant as well as the index form. This book provides up-to-date information on metric (i.e. Killing, homothetic and conformal), connection (i.e. affine, conformal and projective), curvature collineations and curvature inheritance symmetries. It is the first-ever attempt to present a comprehensive account of a very large number of papers on symmetries of spacetimes and Riemannian manifolds. An attempt has been made to present the Lie group/algebra structures of symmetry vectors, their kinematics/dynamics, compact hypersurfaces (dealing with the initial value problem in general relativity) and lightlike hypersurfaces. This book also contains the latest information on symmetries of Kaehler, contact and globally framed manifolds. Audience: Graduate students, post-doctoral students and faculty interested in differential geometry and/or general relativity. Front Matter....Pages i-x Preliminaries....Pages 1-9 Semi-Riemannian Manifolds and Hypersurfaces....Pages 10-35 Lie Derivatives and Symmetry Groups....Pages 36-55 Spacetimes of General Relativity....Pages 56-78 Killing and Affine Killing Vector Fields....Pages 79-102 Homothetic and Conformal Symmetries....Pages 103-133 Connection and Curvature Symmetries....Pages 134-155 Symmetry Inheritance....Pages 156-172 Symmetries of Some Geometric Structures....Pages 173-192 Back Matter....Pages 193-217