There is not much to say about the quality of the text (and the book). As readers mentioned, this is a first-class textbook, and is probably the best for self-study on this subject (Loring Tu's book is also great, but feels a bit "shallow" in terms of examples and exercises). However, my copy already had to be glued back together, even after careful use. [This is a Springer problem, and I will not take any stars away...] Content-wise, it is actually quite similar to Spivak's Comprehensive Intro to Differential Geometry, Vol. 1 -- It's all about manifolds, without specializing on any particular structure. However, it is definitely slower in pace and gentler. Spivak definitely has a way of introducing details that are interesting but difficult and not necessary for a first-timer to go through. And with Spivak, there are still pages I've had to read and re-read just to understand the point, but Lee's text is gentle enough that I usually at least appreciate what he's trying to do, even if I miss the exact reasoning or details on first read. I suppose it's in part because I am reading it after Spivak. Nevertheless, Lee is the rare mathematician who really tries to make sure that students don't get lost. Cynically, for the reason that the text "hand holds" too much, professional differential geometers are unlikely to rate it particularly highly -- mathematicians want texts to be sleek and laconic -- not a word more than what is logically necessary but indifferent to the student seeing the topic for the first time (like any of the three Rudin texts, for example). It's great if a) you have a great professor or b) you happen to be extremely smart. But for most people engaged in self-study, it quickly leads them to give up.
Some say this text book is too verbose, compared to Spivak or Tu. However, there is a good deal of material here that the other two texts don't talk about. For example, there is a good deal of homology theory and algebraic topology, culminating with a long, though very comprehensible proof of de Rham's theorem. Moreover, this is the only one of these texts to deal with Stokes' theorem when there are sharp "corners" involved (which is quite often) instead of brushing it under the rug.
However, the relative lack of focus does make the text harder to navigate than other introductory texts.
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The quality of the binding is unacceptable.
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Livro muito bom.
Livro com um conteúdo muito bom.
Reviewed in Brazil on April 13, 2024
Dr. Wolfgang Klein
Lee's books on smooth and Riemannian manifolds
Reviewed in Germany on March 27, 2021Customer101
Best introduction to smooth manifold
Reviewed in India on April 10, 2018