Differential Geometry

Excellent, but Challenging, Introduction

I strongly recommend this book to anyone looking for an introduction to differential geometry. This book restricts its coverage to curves and surfaces in three dimensional Euclidean space, which is highly appropriate for a first book on the subject. Beyond that, nothing is held back. This book includes a self-contained introduction to tensorial methods in chapter two, and tensors are used heavily in the remainder of the book, which makes this book much more suitable for anyone interested in studying general relativity than a book that tries to limp through the same subject matter using only vector methods. In fact, all of the basic elements that are necessary for the study of general relativity are introduced in this book and in the simplest possible setting. This book includes exactly 99 figures and a large number of examples which are extremely helpful in understanding the material and as other reviewers have remarked has numerous exercises with full solutions in the back of the book. There is also a collection of formulae at the end which makes for a good review and enhances the book's usefulness as a reference. The definitions are explicit and the proofs are quite clear. However, the proofs do make references to the theory of differential equations and to results in complex variable theory in a couple of places. Downsides? While the exposition is excellent, it is a bit terse. Towards the end, there is a lot of flipping back to look at referenced earlier formulas. In addition, small steps are omitted from many derivations. Also, there is a section on the Bergman metric that seemed completely tangential to the rest of the material in the book. Here's a breakdown of the contents: Chapter 1 is preliminaries. It provides a quick review of vector methods and fixes notation. Chapter 2 is the theory of curves in the three dimensions. Topics include: arc length, the tangent vector, the principal normal vector, curvature, binormal vector, torsion, Frenet's formulas, sp...