Summary and Info
I have in mind the following reader: someone who will do what it takes to learn Differential Geometry. The title of this book is not 'Differential Geometry,' but 'Intro to Smooth Manifolds;' a title I think is very appropriate. In this book, you will learn all the essential tools of smooth manifolds but it stops short of embarking in a bona fide study of Differential Geometry; which is the study of manifolds plus some extra structure (be it Riemannian metric, Group or Symplectic structure, etc). I should note, however, that it does cover elementary notions of Riemannian metrics and a fair amount of Lie Groups. At first I found it annoying that I had to work through over 500 pages of dense mathematics before I could study what I really had my heart set on: Riemannian Geometry. But, having read Lee's book cover to cover, I am glad that I waited and developed all the necessary tools. Lee assumes the reader is well prepared, i.e. has had rigorous courses in Multivariable Analysis especially up to Inverse Function Theorem at the level of, say `baby Rudin' (but, Lee does prove this is complete detail), Group theory, Linear Algebra, and Topology. In my opinion, all of these are necessary to really understand the subject. I would advice anyone who will work through Lee's tome to pick up a slimmer, more concise book to stay relatively grounded. My personal favorites are: Janich's `Vector Analysis' (I can't recommend this enough!), Barden's `Intro to Differentiable Manifolds,' Janich and Brocker's `Differential Topology' (hands down, the best pictures!), Milnor's `Topology from a Differentiable Viewpoint.' ** Merits **-Pedagogical, motivational, student friendly (Excellent Index!), lots of details-Moves slow, takes its time developing basics with lots of pictures and heuristic arguments -Lots of worked out examples!-Very good selection of problems-Very useful appendix on Topology, Analysis, and Linear Algebra (A must read as the highlights of the subjects are conveyed with only the useful proofs thrown in)-Prepares one for advanced books in Differential Geometry, i.e. Riemannian Geometry, Differential Topology, etc. -The entire book can be covered in a semester and a half, leaving time to cover most of Lee's Riemannian geometry book.** Simultaneous Merits, Stumbling blocks, and/or Distractions **-Too much information for a first reading-Too wordy (overly detailed in proofs)-Subjects are introduced at the moment tools are available, not in their own separate chapters-Not clear how chapters are interdependent (however, research mathematics is not artificially divided so it's refreshing to read a book that embraces this)** Faults/Disadvantages **-Lots of typos, so be sure to download the list of errata from the authors webpage -Style not for everyone; some readers will prefer more reserved, concise treatments. To this end, I can recommend Warner's `Foundations of Differentiable Manifolds and Lie Groups'-Need to look elsewhere for Riemannian geometry, i.e. there is no mention of a connection or curvature-Not useful as a reference (unless, of course, you worked through it cover to cover and a have a feel for when things were introduced. However, the index is excellent)Please have a look at the reviews by Mr. Raleigh and "math reader." I agree whole-heartedly with their assessment of Lee. They also talk about some aspects that I do not repeat =)** Conclusion **A five star scholarly work! Also, it is incredibly addicting and FUN to read, work from, and learn!