Pure and Applied Mathematics, Marcel Dekker. 202. New York, NY: Marcel Dekker. xiv, 635 p. $ 175.00 (1996).
This is a fully revised and updated second edition of the excellent book of the first two authors [Global Lorentzian geometry, Marcel Dekker (New York 1981; Zbl 0462.53001)]. It continues the mathematical theory of Lorentzian geometry from the viewpoint of global differential geometry. The enormous interest for spacetime differential geometry, especially with respect to its applications in general relativity, has prompted the authors to add new material reflecting the best achievements in the field. The book contains the prefaces for both editions, a list of figures, 14 chapters, 2 appendices, references, a list of symbols and a subject index.
In the first chapter, the authors review the main properties of the distance function and then compare and contrast the corresponding results for the Lorentzian distance function. The second chapter is concerned with the study of connections on semi-Riemannian manifolds ending with a description of the Einstein equations. The causality theory of spacetimes is discussed in Chapter 3. Here there are also presented results on two-dimensional spacetimes, on the second fundamental form of a nondegenerate submanifold and on warped products. In Chapter 4 the Lorentzian distance is studied in detail along with its relations with the causal structure of the given spacetime. Important examples of spacetimes are presented in Chapter 5: Minkowski spacetime, Schwarzschild spacetimes, Kerr spacetimes and Robertson-Walker spacetimes.
In chapter 6 geodesic and metric completeness for Lorentz manifolds and extensions of spacetimes are studied.In particular, local extensions are related to curvature singularities. Chapter 7 deals with stability of completeness and incompleteness for spacetimes. In Chapter 8 maximal geodesics and causally disconnected spacetimes are studied. The importance of the cut locus in Riemannian geometry has influenced the authors to discuss in Chapter 9 the analogous concepts and results for timelike and null geodesics in spacetimes. The Morse index theory on Lorentz manifolds is presented in Chapter 10. The Lorentzian analogous of both the Bonnet-Myers diameter theorem and the Hadamard-Cartan theorem from global Riemannian geometry are obtained in Chapter 11. The main purpose of Chapter 12 is to establish several singularity (i.e., incompleteness) theorems. The last two chapters are concerned with gravitational plane wave spacetimes and the splitting problem in global Lorentzian geometry.
I think the present book will be a most valuable reference for anyone interested in global Lorentzian geometry. I would also like to add that the selfcontained character of this book and the excellent organization of the material make it a perfect source for a graduate course.