Modern Differential Geometry for Physicists (2nd Edition) (World Scientific Lecture Notes in Physics) [Paperback]

C. J. Isham (Author)

Book Description

ISBN-10: 9810235623 | ISBN-13: 978-9810235628 | Publication Date: March 1999 | Edition: 2

These lecture notes are the content of an introductory course on modern, co-ordinate-free differential geometry which is taken by first-year theoretical physics PhD students, or by students attending the one-year MSc course, "Fundamental Fields and Forces" at Imperial College. The book is concerned entirely with mathematics proper, although the emphasis and detailed topics have been chosen bearing in mind the way in which differential geometry is applied to modern theoretical physics. This includes not only the traditional area of general relativity but also the theory of Yang-Mills fields, nonlinear sigma models and other types of nonlinear field systems that feature in modern quantum field theory. This edition of the text contains an additional chapter that introduces some of the basic ideas of general topology needed in differential geometry. A number of small corrections and additions have also been made. The volume is divided into four parts. The first provides an introduction to general topology, the second covers introductory co-ordinate-free differential geometry, the third examines geometrical aspects of the theory of Lie groups and Lie group actions on manifolds, and the fourth provides an introduction to the theory of fibre bundles. In the introduction to differential geometry the author lays considerable stress on the basic ideas of "tangent space structure", which he develops from several different points of view - some geometrical, others more algebraic.

Editorial Reviews

Review

"The result is a book which provides a rapid initiation to the material in question with care and sufficient detail to allow the reader to emerge with a genuine familiarity with the foundations of these subjects." Mathematical Reviews "This book is carefully written, and attention is paid to rigor and relevant details ... The key notions are discussed with great care and from many points of view, which attenuates the shock of the formalism." Mathematical Reviews

Product Details

• Paperback: 250 pages
• Publisher: World Scientific Publishing Company; 2 edition (March 1999)
• Language: English
• ISBN-10: 9810235623
• ISBN-13: 978-9810235628
• Product Dimensions: 8.5 x 6 x 0.6 inches
• Shipping Weight: 1.1 pounds (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #891,329 in Books (See Top 100 in Books)

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40 of 42 people found the following review helpful:

4.0 out of 5 stars Good, with problems, January 1, 2006

By 

Alan Algustyniak (Los Angeles area) - See all my reviews
(REAL NAME)   

This review is from: Modern Differential Geometry for Physicists (2nd Edition) (World Scientific Lecture Notes in Physics) (Paperback)

Wow! What a great Table of Contents. It has all the stuff I've been wanting to learn about. So I bought the book in spite of seeing only one review of it. After one day, I'm now only at page 26, but I already have read enough to make some comments about it.
The main point about this book is that it is, as the author specifically states, LECTURE NOTES, not, I repeat, not a textbook. What are the implications of this (outside of a somewhat more chatty style than a textbook)? ["chatty" isn't quite what I mean; "smooth" might be a better word'] There are two which are noticable to me. 1) A lot of math knowledge is taken for granted. 2) It has a somewhat sloppy style to it.

Regarding point one, make sure you have a lot of math under your belt before picking up this book. By page 18 the author uses these terms without defining them: Differentiable Manifold, semigroup, Riemannian Metric, Topological Space, Hilbert Space, the "<v,v>" notation, vector space, and Boolean Algebra. Fortunately for me, I have a fairly extensive math education, and self-studied Functional Analysis, so I wasn't thrown for a loop; but for many others -- brace yourselves!

Regarding point two, Here are two examples:
1) Here is a quote: "The collection of all open sets in any metric space is called the topology associated with the space." Sounds like a definition to me! Fortunately the author gives a (sloppy) definition a few lines later. By the way, the only thing the reader learns about what an 'open set' is, is that it contains none of its boundary points. All the topology books I have read define open sets to be those in the topology. This is another point of confusion for the reader. In fact, points of confusion abound in that portion of the book.
2) On page, 17, trying somewhat haphazardly to explain the concept of a neighborhood, the author defines N as "N := {N(x) | x is an element of X}" This is already a little disconcerting: x is already understood to be an element of X. So he is saying that N is defined as N(x) (which he defines to be a collection of subsets of X). This is all he has to say on the matter until, on page 26, he writes "each N, an element of N(x)". Now N isn't both N(x) and an element of N(x). This is a point which the author does not clear up. He then starts using N all over the place, yet the reader isn't sure of what he's refering to.

A couple of other things:
-When he defines terms, they is not highlighted, and are embedded in a sentence, making it difficult to find them later.
- The index is pitifully small. Typical for English texts, I know; but this *is* the 3rd millinium!

On the other hand, I have good things to say about the book, too.
I like his style of writing. If it were just more precise, it would be fine for me. I like it better than the normal higher math texts, which tend to be too laconic for me. Notice that I make a distinction between the somewhat chatty style, which I like, and the sloppiness, which is confusing. One can be chatty, yet clear. So far, the undefined math terms which I listed above were not central to the text; and one would not miss much by just reading past them. The author includes many 'comments' sections throughout the book. These are wonderful so far. They are full of comments and examples which really clear up a lot of points. His examples are very good, too, although he is very terse in stating them. The paperback is nice looking. The paper, font, etc. make for easy reading (except for the sub/super-script font, which is too small for me).

To wrap this review up, I had already pretty much learned the stuff covered in the book so far, but judging from what I have read, I will be able to learn a lot from the rest of it; and, unlike some other math books I have studied, the experience won't be too painful.
p.s. See other reviews of it on the UK Amazon site.

19 of 22 people found the following review helpful:

5.0 out of 5 stars Very readable presentation of diff. geometry, August 17, 2000

By A Customer

This review is from: Modern Differential Geometry for Physicists (2nd Edition) (World Scientific Lecture Notes in Physics) (Paperback)

I have found Isham's treatment of differential geometry very clear, while maintaining quite an abstract nature. Isham takes care to motivate his definitions and include comments where comments are due. No problems are included but the book sometimes omits the simpler results and lets you work them out by yourself. A very readable introduction indeed.