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4.0 out of 5 stars An Excursion into the Realm of Differential Geometry, 29 Feb. 2004
This review is from: Elementary Differential Geometry (Hardcover)
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.
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