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Elementary Differential Geometry, Revised 2nd Edition, Second Edition Hardcover – April 10, 2006

ISBN-13: 978-0120887354 ISBN-10: 0120887355 Edition: 2nd
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Book Description

Includes fully updated computer commands in line with the latest software

About the Author

Barrett O'Neill is currently a Professor in the Department of Mathematics at the University of California, Los Angeles. He has written two other books in advanced mathematics.

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Product Details

  • Hardcover: 520 pages
  • Publisher: Academic Press; 2 edition (April 10, 2006)
  • Language: English
  • ISBN-10: 0120887355
  • ISBN-13: 978-0120887354
  • Product Dimensions: 6.1 x 1.1 x 9.2 inches
  • Shipping Weight: 2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (12 customer reviews)
  • Amazon Best Sellers Rank: #837,418 in Books (See Top 100 in Books)

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4.6 out of 5 stars
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53 of 55 people found the following review helpful By A. Ali on October 17, 2000
Format: Hardcover
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.
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31 of 33 people found the following review helpful By Mark Arjomandi on November 30, 2003
Format: 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 differential geometry. The class --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 term however, I decided to change direction and as time went on, I drifted more and more towards geometry as the field of further concentration. The original second edition (from 1997) contained numerous typos, but luckily, the revised 2006 issue takes care of these and also streamlines the section numbering formats which had made the referencing and following through with the material a bit cumbersome. As some of the other reviewers have mentioned, the emphasis here is on the low (= 2 and 3) dimensional geometry, formulated in the language of differential forms (Cartan's early 20th century approach).

Within the eight chapters of the book (seven in the 1966 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 is on the study of the shape operators and normal and Gaussian curvatures, where also some useful computational examples have been presented.
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26 of 29 people found the following review helpful By Rehan Dost on July 10, 2005
Format: Hardcover
If you are looking for abstraction with little in the way of intuition I suggest Conlan " differential manifolds"

If you are an applied mathematician or physicist this book is for you.

I have always beleived that to truly grasp mathematics one must be provided with a reason for WHY things are the way they are and WHAT IDEAS the expression must express. This is best done with examples and exercises.

I digress.

The book restricts is exposition to two and three dimensions. Some of the topics can readily be bootstrapped to higher dimensions.

The book starts with basic ideas of curve, directional derivative and tangent vector in Euclidean space with a sprinkling of differential forms to wet the appetite.

It then moves into the notion of frame fields along curves resulting in the Frenet formulas which express how the frame fields change along the curve. These are expressed in terms of the frame field themselves giving ideas of curvature and torsion.

The book then abstracts these concepts to show how we can talk about change of frame fields along arbritrary directions not just along the curve. The tools used to do this are the covariant derivative and connection forms which can then be used to develop connection equations ( abstracted analogue of frenet formulas ) and then the cartan structural equations.

The book talks about isometries and defines euclidean geometry as those properties preserved by isometries. It then abstracts once again to surfaces in R3 using patches and appropriate conditions on the overlap without introducing manifolds although these are briefly mentioned later.

We then see how calculus in euclidean space can be adapted to surfaces using these patches.
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Elementary Differential Geometry, Revised 2nd Edition, Second Edition
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