Semi-Riemannian Geometry

Preface

Physics texts on general relativity usually devote several chapters to an overview of semi‐Riemannian geometry. Of necessity, the treatment is cursory, covering only the essential elements and typically omitting proofs of theorems. For physics students wanting greater mathematical rigor, there are surprisingly few options. Modern mathematical treatments of semi‐Riemannian geometry require grounding in the theory of curves and surfaces, smooth manifolds, and Riemannian geometry. There are numerous books on these topics, several of which are included in Further Reading. Some of them provide a limited amount of material on semi‐Riemannian geometry, but there is really only one mathematics text currently available that is devoted to semi‐Riemannian geometry and geared toward general relativity, namely, Semi‐Riemannian Geometry: With Applications to Relativity by Barrett O'Neill. This is a classic, but it is pitched at an advanced level, making it of limited value to the beginner. I wrote the present book with the aim of filling this void in the literature.

There are three parts to the book. Part I and the Appendices present background material on linear algebra, multilinear algebra, abstract algebra, topology, and real analysis. The aim is to make the book as self‐contained as possible. Part II discusses aspects of the classical theory of curves and surfaces, but differs from most other expositions in that Lorentz as well as Euclidean signatures are discussed. Part III ...