Elementary Differential Geometry

Rating:
Summary: Solid and Modern Introduction
Review: I worked through the first edition of this book some years back. After finishing this book I was ready for more abstract treatments of Riemannian Geometry. For example, having seen covariant derivatives on 2-surfaces embedded in R^3 motivates the abstract definition of connections on manifolds.

Chapter 1 is a decent introduction to pullbacks and pushforwards of differntial forms and tangent vectors respectively. In fact, all the subsequent geometry is based on pullbacks and pushforwards.This itself motivates the more abstract definition of a differentiable manifold with its coordinate charts. True,tangent vectors are not described in the most abstract fashion (e.g. as derivations on the algebra of functions) but this is not appropriate for a first course.

Chapter 2 describes the language of frame field and connection forms and derives the Frenet-Serret equations in terms of moving frames and structure equations. We associate this with the methods of Elie Cartan, who used moving frames in a systematic manner.

Chapter 3 deals with isometries; frankly speaking I never understood the raison d'etre for such a long chapter on such a topic.

Chapter 4 discusses coordinate patches. Again, this is thoroughly modern, and you won't find this in Struik or Kreyszig. The idea of piecing together coordinate patches to get geometric or topological information is a twentieth-century conception.

Chapter 5 introduces the Shape Operator, which is subsequently used in Chapter 6 to derive the equations of surface theory. This is really moving frames again, in another guise.

Chapter 7 finally tries to put this in a more abstract setting by defining abstract surfaces with an intrinsically defined covariant derivative.Holonomy and the Gauss-Bonnet theorem are discussed.

After reading this book, one would be equipped to handle do Carmo's book on Riemannian geometry, or O'Neill's book on Semi-Riemanninan geometry, or the more recent book by Lee, again on Riemannian geometry.

Rating:
Summary: Good low dimensional calculation
Review: It's easy to read with enough examples. Suitable for self study after your advanced calculus (inverse function thm/implicit function thm should be covered here) and linear algebra classes. Tons of exercises will help you familiarize yourself with the calculation in low dimension. (Do I love the exercises on minimal surfaces and surfaces of revolution in chapter 5 and 6!) Most of them are workable. This is the strength of the book. Since the author limits the material to low dimensions, some definitions are a bit misleading, such as the definition of exterior derivative of 1-form in chapter 4, where another term to be added happens to be zero. I think there is a big gap in style and level of difficulty between this book and author's "Semi-Riemannian Geometry: With Applications To Relativity".

After this book, probably you want to read Hicks' "Notes on differential geometry", if you can find a copy in some lib. Darling's "Differential Forms and Connections" is also highly recommended. It is modern but not much topological stuff.
Company it with Warner's "Foundations of differentiable manifolds and Lie groups" for topology also much higher algebra.

Rating:
Summary: An Excursion into the Realm of Differential Geometry
Review: My first encounter with this book was during the academic year of 2000-2001, when it was used as the main text for an upper division course on differnetial geometry, at one of the University of California campuses . The class for me --taught by a distinguished scholar-- was only meant to be a brief excursion into the realm of continuous math, beyond analysis and topology. After finishing the class however, I decided to change direction and as time went on, I drifted more and more towards geometry as the field of further concentration. Before we proceed further, let me note that one main complaint that's rather well-known about this text is the issue of numerous typo's therein. What many may not know however is that the first edition from 1966 does not contain any noticeable typo's, unfortunately somehow all of them found their way in the 1997's second edition. This is very likely because of careless typesetting, but one good news is that many of these are noted on the errata sheet available from the author's UCLA web site. Moreover, the book uses a cumbersome section numbering format, and to make things worse, in quite a few places the reader is referred to one or more previous sections. This serves to disrupt the flow of reading by taking up some time and effort to locate the correct previous page number which is being referenced.

Within the eight chapters of the book (seven chapters in the first edition), the reader is first introduced to some preliminaries such as tangent vectors, directional derivatives, and differential forms. In chapter two, the author presents the Frenet frame formulas, covariant derivatives, connection forms, and Cartan's structural equations, which are generalizations of the Frenet frame formulas for surfaces. In chapters three and four, there is a healthy dose of Euclidean geometry and calculus on surfaces. In chapter five, discussion shifts to the study of the shape operators and normal and Gaussian curvatures, where also some useful computational examples have been presented. Geometry of surfaces is the subject of chapter six, where the crucial Gauss' egregium theorem and some global theorems are also discussed, and in chapter seven students are introduced to the basics of the Riemannian geometry, culminating in the famous Gauss-Bonnet theorem. In chapter eight (which is highly topological), the concepts of geodesics, complete surfaces, covering spaces, Jacobi fields, conjugate points, and a couple of constant curvature theorems for surfaces are explored. The appendices include help on using popular computer algebra systems, and another appendix providing solutions to most of the odd-numbered exercises in the book.

Again, looking on the downside, the book lacks a discussion of several essential tools, for example, the Schwarz-Christoffel symbols, tensors, and Lie derivatives, as well as some other important topics such as the first and second fundamental forms, and parallel translation, which only show up in the exercises. Then again, perhaps to keep the level of exposition elemantary and the size limited to less than 500 pages, Dr. O'Neill has preferred to skip some topics. One remedy is to back this text up with Manfredo Do Carmo's 1976 classic, which is mathematically more rigorous, and covers more of the above-mentioned topics (be aware though that Do Carmo is less accessible for the beginning students). Afterwards, one can certainly continue the study of the essentials by reading other advanced books such as Barrett O'Neill's (obscure) graduate-level 1983 treatise on Applications of the Semi-Riemannian Geometry to Relativity, or William Boothby's "An Introduction to Differentiable Manifolds and Riemannian Geometry". One other underrated source which is worthwhile to look into is Richard W. Sharpe's "Differential Geometry: Cartan's Generalization of Klein's Erlangen Program", from the Springer-Verlag GTM series.