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!
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Jordan Causon
Product came in perfect condition. Thank you.
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