This book is an exposition of semi-Riemannian geometry ---also called pseudo-Riemannian geometry--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry.
For many years these two geometries have developed almost independently: Riemannian geometry formulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation, devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest.
After establishing the requisite language of manifolds and tensors (Chs.1 and 2), the plan of the book is to develop the foundations of semi-Riemannian geometry in the simplest way and without regard to signature, allowing the Riemannina and Lorentz cases to appear as needed (Chs.3-5 and 7). Then in the latter half of the book two threads are followed. One uses the notion of isometry to develop algebraic aspects of semi-Riemannian geometry: manifolds of constant curvature, symmetric spaces, and homogeneous spaces (Chs.8,9,11); the introductions to these chapters will give a more detailed description of their contents. The other thread applies Lorentz geometry to special and general relativity (Chs.6,12,13).
The fact that relativity theory is expressed in terms of Lorentz geometry is lucky for geometers, who can thus penetrate surprisingly quickly into cosmology ( redshift, expanding universe, and big bang) and, a topic no less interesting geometrically, the gravitation of a single star (perihelion advance, bending of light, and black holes). The tendency of the spacetimes in Chapters 12 and 13 to have singularities (big bang and black holes) is accounted for in abstract Lorentz terms by two theorems, due respectively to Stephen Hawking and Roger Penrose: these are the goals of Chapter 14.
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. In particular, the definition of tensor I have adopted converts almost automatically into the classical coordinate formulation. A number of key proofs are given classical notation. This attitude is only reasonable in view of the vast literature in each style.
The basic prerequisites for the book are modest: a good working knowledge of multivariable differential calculus, a firm belief in the existence and uniqueness theorems of ordinary differential equations, and an acquaintance with the fundamentals of point set topology and algebra. Later on, a knowledge of fundamental groups, covering spaces, and Lie groups is required; the necessary background in these topics is outlined briefly in Appendixes A and B.
A college course in physics (particularly Newtonian mechanics) is required, not to read this book, but to appreciate the transformation and unification of Newtonian concepts effected by Einstein's relativistic geometry and the remarkable way the old and new theories--so different at base--reach approximate agreement on, say, the running of the solar system (Appendix C versus Chapter 13).