Differential Geometry (Dover Books on Mathematics) First Edition

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, spherical images of curves, the canonical representation of curves, orders of contact between curves, natural equations for curves, involutes and evolutes, and more.Chapter 3 introduces surface theory and covers the first fundamental form, normals to surfaces, and an introduction to tensorial methods. This introduction is good, self-contained, and covers only the tensor calculus that is required for the rest of the book. Tensors are presented using index notation rather than the more modern -- and for me at least usually less clear -- abstact notation. The Einstein summation convention is introduced immediately and used throughout except in formulas where it is explicitly suspended.Chapter 4 covers the second fundamental form, gaussian and mean curvature for a surface, Gauss' Theorema Egregium, and Christoffel symbols.Chapter 5 is about geodesics and also covers the Gauss-Bonnet theorem.Chapter 6 studies mappings and provides good coverage of various types of mappings of a sphere into a plane such as conformal and equiareal. It also covers conformal mappings of three space.Chapter 7 discusses absolute differentiation and parallel transport. It also has a section on connections in general. Absolutely key material for understanding general relativity.Chapter 8 tackles special surfaces such as minimal surfaces, modular surfaces of analytic fucntions of one complex variable, and surfaces of constant gaussian curvature.This book absolutely requires a strong background in multivariable calculus and differential equations. In addition, some exposure to complex variables is recommended.I strongly recommend this book for any scientist or engineer looking for an introduction to differential geometry. If this book proves to be too much, then I'd suggest looking at a book that makes ues of only vector methods for some additional background before returning to this book. Finally, the price is hard to beat!

This is a surprisingly modern work when one considers the 1959 publication date. It covers both Riemannian geometry and covariant differentiation, as well as the classical differential geometry of embedded surfaces.The first two chapters of "Differential Geometry", by Erwin Kreyszig, present the classical differential geometry theory of curves, much of which is reminiscent of the works of Darboux around about 1890. Then there is a chapter on tensor calculus in the context of Riemannian geometry. Chapter 4 is about the second fundamental form and the mean and Gaußian curvatures, including the Riemannian curvature and Christoffel symbols. Chapter 5 is about geodesics in the Riemannian geometry context, which is less general than the fully general affine connection context. Then following a chapter on maps, there is Chapter 7 on covariant derivatives. Although it appears at first that this is a metric-free treatment, the metric is introduced after several pages as if it was always there. This can lead to confusion because many formulas for covariant derivatives which are valid for the Riemannian metric context are not valid in the general affine connection context.Finally there is a big Chapter 8 containing examples of special geometries. This books has lots of practical examples, and lots of problems and answers to problems. There is also a helpful 15-page summary of formulas at the end of the book.PS. 2013-6-28. After making the above comments about the 1959 Kreyszig book yesterday, I noticed that the 1959 Willmore book "An Introduction to Differential Geometry" is very much more modern than the Kreyszig book. For example, the Willmore book presents compactness issues regarding geodesics, various global topology results, general affine connections without the Riemannian metric, torsion of connections, connection forms and fibre bundles. These more modern topics are effectively absent from the 1959 Kreyszig book. Kreyszig essentially assumed the Riemannian metric throughout, whereas Willmore presented the first 225 pages without the metric, and then presented how the situation changes when you do have a metric.

This is a good book to go through to get an understanding of differential geometry. It has all the things you need to study. I had hoped that it would be a bit more general, but the book heavily focuses on 2-dimensional surfaces. I like that it has problems with solutions at the back of the book. This makes it much better suited for self-study. I also thought it had fairly clear explanations overall. There were a couple of times I thought the presentation could have been clearer, however, and some of the problems are less enlightening than I would have hoped. That is, a few problems seemed purely for calculation without really giving any deeper insight into anything, but the majority of problems were good and helped cement understanding.