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Semi-Riemannian geometry. With applications to relativity. (English) Zbl 0531.53051
Pure and Applied Mathematics, 103. New York-London etc.: Academic Press. xiii, 468 p. $ 45.00 (1983).
The principal object of this book is to give an introduction to semi-Riemannian geometry and some important applications to general relativity. A semi-Riemannian manifold (also called pseudo-Riemannian manifold by other authors) is a smooth manifold \(M\) and a \(C^{\infty}\) symmetric nondegenerate tensor field \(g\) of type \((0,2)\) with constant index \(d\). If \(d=0\), \((M,g)\) is a Riemannian manifold. If \(d=1\) and \(n\geq 2\), \((M,g)\) is a Lorentz manifold. The book contains fourteen chapters and three appendices.
In the first two chapters the language of manifolds and tensors is established. The next chapter is devoted to covariant differentiation, geodesics, sectional curvature, some curvature operators and local isometries. In chapter 4 we find one of the first systematic expositions of semi-Riemannian submanifolds in a book. A submanifold \(N\) of a semi-Riemannian manifold \((M,g)\) is a semi-Riemannian submanifold iff each of its tangent planes \(T_pN\) is a nondegenerate subspace in the ambient tangent space. In this case the index of \(T_pN\) is the same for all \(p\). The second fundamental form is defined and the Gauss-Codazzi equations are established. As an application one studies semi-Riemannian hypersurfaces in a semi-Euclidean space \(\mathbb R^{n+1}\).
The fifth chapter is a comparative study of Riemannian and Lorentz manifolds. A useful analogy appears between timelike geodesics of a Lorentz manifold and geodesics in a Riemannian manifold. But, for a semi-Riemannian manifold, in particular for a Lorentz manifold, the completeness is a more subtle notion than in the Riemannian case, since the Hopf-Rinow theorem has no satisfactory generalization. For a detailed study the book of J. K. Beem and P. E. Ehrlich [Global Lorentzian geometry. New York etc.: Marcel Dekker (1981; Zbl 0462.53001)] is recommended. Certain constructions of new semi-Riemannian manifolds are made in chapter 7 by using some properties of covering manifolds. In chapters 8,9 and 11 the author uses the notion of isometry to develop algebraic aspects of semi-Riemannian geometry: manifolds of constant curvature, symmetric spaces and homogeneous spaces. He gives a list of all complete, simply connected manifolds of constant curvature. Completeness of a compact homogeneous semi-Riemannian manifold is proved. In chapter 10 “Calculus of variations‘” useful formulas for the first and second variations of arc length and the index form are examined.
In chapters 6, 12, 13 and 14 the author applies Lorentz geometry to special and general relativity. Chapter 12 is a brief account of the fundamentals of general relativity and some information about the origin and development of the universe. It is interesting to observe that the geometric concept of “warped product” is used to build quite simple cosmological models whose properties have a reasonable chance of being physically realistic (Schwarzschild, Robertson-Walker, Friedman, Einstein-de Sitter, et al.). Chapter 13 is devoted to the geometry of Schwarzschild space-time. In the last chapter it is shown that the property of cosmological models of having singularities is an essential one. To prove this for large classes of space-times, special and nonstandard investigations are needed, in which geometric methods must be combined with topological ones. The author proves two remarkable theorems – one, due to Penrose, motivated by black hole singularities, the other, due to Hawking, motivated by cosmological (e.g., big bang) singularities.
For more results on singularities the book of S. W. Hawking and G. F. R. Ellis [The large scale structure of space-time. London: Cambridge University Press (1973; Zbl 0265.53054)] is recommended. In Appendices A and B the necessary background from the theory of fundamental groups, covering spaces and Lie groups, is presented. A review on Newtonian gravitation is given in Appendix C.
One has to note that this book contains a lot of recent results from Lorentz geometry with surprisingly simple proofs given by the author. The book is clear and very carefully organized. Examples are everywhere dense and illustrate the theory very well. At the end of each chapter there are exercises of varying difficulty. The general approach of the book is coordinate-free; however, coordinates are not neglected. Typically, geometric objects are defined invariantly and then described in terms of coordinates. The didactical qualities of the book are very well combined with a high level of scientific rigour. The scope of the contents and the clarity of exposition make the book a useful reference in the geometry of semi-Riemannian manifolds (with index \(d\neq 0)\). Also, this excellent book will be of great interest to both geometers and physicists working in general relativity and gravitation.

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53B50 Applications of local differential geometry to the sciences
83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
53C80 Applications of global differential geometry to the sciences