[.ca] Introduction to Smooth Manifolds (ISBN 0387954481)

Excellent, lucid book on manifolds:
Topics are explained with exceptional clarity; portions of the book are well tied together; and the order of exposition flows very well. Lie groups are introduced quite early on, but their full power is not revealed until later in the book. I can't laud this book enough. I had a firm, well-developed basis of differential geometry after reading through this book for a course. The excersises are illuminating, as are the examples. Theorems and their proofs are clearly labeled. The motivational explanations prefacing theorems do an excellent job of conveying the intuition behind ideas. I would recommend this book over Boothby any day. I haven't read Spivak, so I can't compare Lee to it, but Lee definitely seemed like an excellent choice for an intro grad class on differential geometry.

get the hardcover!!:
I just finished a 20-week course from this book. It is well-written, with a healthy number of examples and many exercises (interspersed throughout the text) and problems (at the end of each chapter). The style is rather informal: this is good for the novice to this subject, which groans under the weight of its own notation. The presentation is well-organized, clear, and accessible. Dr. Lee maintains current errata for the book (some did make it into the problems unfortunately) at his website. One thing I might suggest is that if you plan to use this book heavily (e.g., for a course rather than for reference or bedtime reading) you should consider investing in the hardcover version is possible. The book is lengthy and the binding tends to split. Mine is still in one piece, but only just. You have to be very gentle with this book to keep it intact.

Extremely well written:
If you want to learn smooth manifold theory then this book is your best choice. Lee's style of writing is exceptionally ... smooth. The style is intended to iluminate the subject without comprimising rigor. It's a "real" math book so there is very little handwaving. I particularly like the organization of the book. The short length of the chapters motivates the learning process and builds motivation (...that's one more chapter under my belt...). There are 20 chapters in total with average length of 20-30 pages. Long chapters in a book can lead one to forget what is being studied. By making the chapters short, Lee helps the reader focus on learning the particular topic. The sequence of the chapters then allows one to envision the whole picture. The appendix contains some basic background from topology, linear algebra, and calculus. If you are not sure about a certain term you will probably find it there. My hats off to Lee. I plan to buy his other books.

Extremely well written:
If you want to learn smooth manifold theory then this book is your best choice. Lee's style of writing is exceptionally ... smooth. The style is intended to iluminate the subject without comprimising rigor. It's a "real" math book so there is very little handwaving. I particularly like the organization of the book. The short length of the chapters motivates the learning process and builds motivation (...that's one more chapter under my belt...). There are 20 chapters in total with average length of 20-30 pages. Long chapters in a book can lead one to forget what is being studied. By making the chapters short, Lee helps the reader focus on learning the particular topic. The sequence of the chapters then allows one to envision the whole picture. The appendix contains some basic background from topology, linear algebra, and calculus. If you are not sure about a certain term you will probably find it there. My hats off to Lee. I plan to buy his other books.