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Title: Introduction to Smooth Manifolds
ISBN: 0387954481
Author:
John M. Lee
Publicate Date: 2002-09-23
Publish: 2002-09-23
List Price: $59.95
Average Customer Rating: 5.0
Format: Paperback
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Amazon Lowest New Price: $40.00
Amazon Lowest Used Price: $43.08
Amazon Merchant Price: $49.08
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| Customer Review: |
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1: The printing is not up to the standard of the writing
By all accounts, this and Dr. Lee's other two books on manifolds are exceptionally well-written. But my copies arrived from Amazon this week, and, unfortunately, Amazon and Springer have decided to replace the crisp offset-printing of earlier printings by lower quality digitally-printed versions, probably as a cost-cutting measure.
If you care about how books look, I'd suggest trying Amazon marketplace or small retailers elsewhere to increase your odds of getting a superior copy from an earlier printing.
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2: The perfect book for your first class on smooth manifolds
This book is an antidote to the more common style of math text. So many math books feel like they were written by mathematicians, which is to say their authors prize being terse over being understandable. Far too many textbooks out there have me jumping on the internet to find explanations and examples of key concepts that absolutely must be understood completely for the sake of even being able to read the next ten pages. Lee spares his readers the trouble by taking things slowly and including all the steps they need to keep up with the discussion.
This was the textbook for my first class on smooth manifolds, and it is worth noting that it was the only class I took that semester in which I learned more from the textbook than from the professor's notes.
If you are a grad student, this book will make your life considerably easier.
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3: Quite a handy reference for geometers
My field lies somewhere at the intersection of algebra, geometry and physics. This is a very handy reference, meaning a few pages accessible and contains most of the basic notions of differentiable manifold and some useful beyond-elementary topics for me. A good book to look into if you can't remember something in details.
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4: Great for self-study
I should say first that I was already familiar with manifold theory before picking up this book. I had already wrestled with some of the definitions, theorems, and whatnot, so I can't necesarily say I was a complete beginner before reading this book. Also, I'm not sure if I can say how great this book would be if you have no idea what a manifold (or tangent space, etc.) is. However, that stuff aside, this is an amazing text. I'm studying this book on my own, and it's great. The concepts are woven throughout the text instead of being lumped into chapters devoted to them (though some people might prefer the latter). Also, they're used to reinforce and build on each other.
As an example, Spivak doesn't treat Lie groups until the second to last chapter. Lee introduces them in the second chapter, uses them as examples throughout the text, builds up the theory of Lie groups as the book goes on, uses Lie groups (and their actions on other manifolds) in developing certain other areas (it really streamlines the development) and ends with a nice big chapter on them. Of course, this is just one example.
Lee developes manifold theory so that it would appeal to a physicist, geometer, algebraist, topologist, etc. Everything gets talked about! This means, however, that he can't treat any one subject in too much detail. For instance, he leaves curvature and other parts of Riemannian geometry to his other Riemannian Geometry text, but it's definitely worth the trade off. This book trashes Spivak. Buy it today!
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5: If we make the assumption that "good book" means a book
that lends itself to self-studying then this is not a good book, but excellent. All complaints reported in other reviews are actually answered in the preface: the book is about the mathematical machinery ordinated under the title smooth manifold theory. It is not a book on Riemannian geometry that's why there is no extensive treatment of metrics or any treatment of connections. Each topic comes up whenever the prerequisite tools have been built and enough motivation can be given, that's why it is a pleasure to read this book. If you like encyclopedic expositions there are plenty of them out there. The appendix contains a compilation of virtually all the results you need from calculus, linear algebra, and topology in order to study smooth manifolds. Perhaps the first time I actually read the appendix of a book (Arnol'd 's books form an exception) It is obvious that the author belongs to that group of people who like to excel in whatever they do, as all his books not only teach you the subject of their titles but also how to write a book if it happens to reach that point in your mathematical career. They are in some sense both books and meta-books on mathematics.
This review is not intended to comment on other reviews, but let us be honest and agree on the fact that an author never faces the danger of being too clear: as to the length and the pace of the book, I wish this book were only one volume of a series from the same author starting with topology and culminating with the interplay of differential geometry and PDE. There is a drawback however, reasonably not anticipated. Most math books are not written to be actually read (aphoristic but true). This one makes an exception and thus the binding proves insufficient quickly. A hardcover version would be convenient. Suggestion for clever math students: learn the stuff from Lee and then pretend you are reading Lang's "introduction"...
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