Reviews (13)

Best DG book out there
This book is rather expensive, but when compared to the other books available, it is not a waste of your money. It has plenty of exercises, many of them with answer or hints in the back of the book, and its exposition is broad, very clear and concise.

It is hard to tell being a math student, but I think anyone with a solid knowledge in multi-variable calculus (Apostol's book would be perfect) or, better yet, who has taken multi-variable analysis course would find this book accessible. One of the advantages of this book is that it is self-contained, so even though it uses, for example, the inverse function theorem (which is something unavoidable for a DG book), it has an appendix on differentiability and continuity which covers this.

The exercises range from easy to very hard, but because of the exposition and of the way the exercises are stated (the tougher ones are many times itemized so that they drive you to the answer) it is rare to find a problem that the reader will not be able to solve upon a little thinking.

The greatest advantage of this book is its clear and well-written exposition. It has very few errors, mostly typographical. It covers a lot of topics and its notation is extremely simple and suggestive, which, believe me, is of great help in a DG book. In short, if you want or have to learn differential geometry, save your time and get this book. As another reader very intelligently put it, there is a reason why this is a classic.

There is a reason why it is a classic.
Before talking about the book itself, let me tell you that I am a mathematician, and when I took a differential geometry course and used do Carmo's book, I already knew I wanted to be a mathematician. So, is this a book for mathematicians? Well, yes, but not exclusively. It is certainly written from a mathematician's point-of-view, and it assumes some maturity on the part of the reader. For instance, the exercises contain very little in the way of drill, and are used to enhance the theory (as pointed out by another reviewer). It seems to me that the author believes that mature readers can provide their own `drill' exercises. So, you won't find many exercises asking for you to find pricipal curvatures for this or that surface, and that other as well; exercises in this book have a theoretical flavor to them. This, of course, makes for some hard exercises, and I do remember spending a lot of time over them, often working together with other students taking the same course. The upside is that we learned the material, and thoroughly. Also, the author provided plenty, plenty of examples. The figures are very well drawn and really allow you to see what is going on - even though these days, with powerful computer packages like Maple, Mathematica, Matlab, and others, any student can provide his/her own pictures. But just because now we can use computers, I wouldn't say the text shows signs of age. It is jus as clear now in its exposition of topics and concepts as it was many years ago. So, even though there are many good alternatives in the market, if I had to teach a course now on this subject, or even better, if I were a student now taking this subject, I would certainly have this book at the top of my list of possible textbooks.

Isbn: 3540411089
Sales Rank: 1050886
Subjects:  1. General    2. Geometry - Differential    3. Mathematics    4. Riemannian Manifolds    5. Science/Mathematics    6. Lie groups    7. Mathematics / Geometry / General    8. Riemannian geometry    9. connections    10. symmetric spaces   

Isbn: 0471157333
Sales Rank: 562840
Subjects:  1. Differential Geometry    2. Geometry - Differential    3. Geometry, Differential    4. Mathematics    5. Science/Mathematics    6. Topology    7. Differential & Riemannian geometry    8. Mathematics / Geometry / Differential   

Reviews (3)

A Gem of Exposition
This book needs no review, as it is so well-known. But reading the first critique about the need for a more intuitive development of the subject, perhaps one viewpoint is worth restating.This book is a model of concise exposition. All of Milnor's works are written in a fashion that "makes it clear", except that sometimes years of thought are really needed to see the essence of the arguments. The writing sets up the framework for understanding, and the reader must work to fill it in. This is what makes it great writing. The reader accepts the truth of the statements, but Milnor does not bludgeon the reader with details which can be filled in by a professional mathematician. Or as many of us understand it, when you can fill in all the details, you have developed geometric intuition and are on the way to a deeper understanding of the subject.

Each chapter of the book is a classic. Chapter 2 on Riemannian geometry gives an overview of the subject which can be used as a basis for teaching a course on the same. When the students can fill in the details, they understand the core of the subject.

The best
When I was just becoming a mathematician, my teachergave me this book, saying "You're not ready for thisyet, but you should have it --- it's the best piece of mathematical exposition there is." Maybe that claim's exaggerated, but I've yet to find one I prefer. Along with Milnor's Lectures on the h-Cobordism Theorem, and hisCharacteristic Classes, this book is a lesson not only in topology (and wonderful topology, too!), but in clear writing as well.

Reviews (1)

The Great American Differential Geometry Book
Michael Spivak begins these five volumes stating his modest aim to write the "Great American Differential Geometry book." He surely has.Instead of listing the numerous subjects Spivak treats clearly and beautifully in these volumes, I'd like to call out the delightful travelogue style in which they are written, using history, anecdotes, and opinion to explain, illuminate, and, when possible, motivate the gleaming modern edifice. Spivak's opinions are sprinkled lightly here and there like easter eggs. How could you not love a math book that uses the subtitle "The Debauch of Indices," or dismisses Eric Temple Bell's history as "supercilious remarks of questionable taste"? Also, don't miss the annotated bibliography in volume 5. The fact that legions of professionals refer to these books in their original *typewritten* format [1st & 2d editions] is a further testament to their quality. The third edition is typeset using TeX and, though beautiful, still manages to retain a little of the quirky typewritten appearance. One quibble: I was disappointed to see that this edition did not use Richard Bassein's bizarre artwork [think 70s psychedelic] for the covers; I admit that this stuff weirded me out originally, but have grown to love it -- where else could I see fuzzy trolls in crowns made from Enneper's minimal surface?

Isbn: 0306109956
Sales Rank: 2954477
Subjects:  1. Differential Geometry    2. Differential Topology    3. Geometry - Differential    4. Geometry, Differential    5. Mathematics    6. Science/Mathematics    7. Topology   

Isbn: 0387904220
Sales Rank: 439448
Subjects:  1. Complex manifolds    2. Differential Geometry    3. General    4. Geometry, Differential    5. Mathematics    6. Science/Mathematics    7. Theory Of Functions    8. Komplexe Mannigfaltigkeit    9. Manifolds    10. Mathematics / Mathematical Analysis   

Isbn: 0387134638
Sales Rank: 1463382
Subjects:  1. Differentiable manifolds    2. Differential forms    3. Geometry - General    4. Mathematics    5. Riemannian Manifolds   

Reviews (3)

Don'twaste your money
This review refers only to the book printing quality not to the contents.

I had purchased some books from Springer in the past (Like Arnold Mathematical Methods of Classical Mechanics, Lang Algebra etc..) and found them beautifully edited: good binding, paper etc..

And to my surprise I was very disappointed with the overall quality of this book, poor binding -glued instead of sewn- bad quality paper -forming waves at the binding spine, etc..

You pay for a quality item, a book you can use for years, and you get a hardbound crap that you can not left open in a table without holding it tight risking to lose the pages after a few days of use in the process.

I find this unacceptable in books costing 60$+. Sadly I find this to occur very often, publishers should be more careful with their printings and custumers should demand a better quality.

Don't waste your money.

A reader.

Good, as long as you have enough background
I read this book at the very beginning of my studying in differential geometry and was striked. The definitions and methods used in this book seemed totally incomprehensible to me. However, after some development in this field, I found that this book is very concise. It is a very good surey on differential geometry but not a good book to start with. Definitions are given from the most "down to bottom" one. It is a very good attitude, yet, if you do not have much background in differential geometry, this book may takes you several days in order to understand the concept of tensor and exterior algebra.