An Introduction to Manifolds (Universitext) 2nd ed. 2011 Edition

When I first began reading the text, I had a difficult time understanding the concepts, but the presentation of the material really laid bare all of the esoteric topics that I hadn't encountered formally before.Loring Tu has done an excellent job of making sure even the uninitiated student can make his/her way through this text, having sprinkled a few easy exercises through the text itself to emphasize the learning and familiarity with definitions, with more difficult exercises at the end (including computations as well as topics that force a student to understand and digest the section immediately preceding the problems). He labels every problem, so a student doesn't wade through pages of text needlessly trying to discover which part of the text will be most useful, but this method allows the student to hone in on the material which is exactly pertinent to that problem. I am by far not the best and brightest student, but I have been able to read the text and given a few hours for each section, complete all exercises throughout the reading and at the end of the section. With many hints and solutions at the end of the textbook, I can be sure I'm not only learning the material, I'm learning it correctly!I would agree with some of the other reviewers that this should be a text every graduate student in mathematics should read. It is not out of the realm of possibilities for a student to read it on his/her own, and the enlightenment gained from the generalizations of multivariate calculus is really a gift to oneself, as well as to any future students the person may have, for they will be able to answer any up-and-coming student's questions with a clarity surpassing any instructor I've personally had, which would have been very helpful as a budding mathematician.

This seems to be a very good book, it is easier than graduate texts I would say at a advanced undergraduate level it covers many topics starting with flat space and calculus on it like R*n and then starts with Manifolds, it even brings a chapter on Lie Groups and Lie Algebras an another on Categories and Functors. But I have not read any of these chapters I immediately went for the last chapter, chapter 7 De Rham Theory, which consist in 6 subchapters: 24-De Rham Cohomology, 25-The Long Exact Sequence in Cohomology, 26-The Mayer-Vietoris Sequence, 27- Homotopy Invariance, 28- Computation of de Rham Cohomology, 29-Proof of Homotopy Invariance. These sections actually taught me HOW TO USE AND CALCULATE COHOMOLOGY GROUPS with the Mayer Vietoris Sequence and for this an only this it is worth it to buy it, here you will find how to calculate the de Rham Cohomology groups for any oriented Riemann surface of whatever genus you want!!!! and this is very important because de Rham's Cohomology groups are very important topological invariants of Manifolds, I am glad I purchased this book and learnt this stuff.

I used this book for a semester long senior undergraduate/masters level class that culminated in Stoke's theorem. I found the material fascinating and thought this book did a good job of being self-contained in developing the basic machinery for integration on manifolds via partitions of unity, while also giving a taste of some interesting related topics: several chapters about Lie groups, immersions/submersions, regular/critical points, and de Rahm cohomology at the end. I especially enjoyed the 5 page section on the category theoretic perspective and the functorial nature of the pullback and pushforward. No complaints really, maybe it could use a few more exercises, but the ones in the book are pretty good. I would have liked discussion of the hodge dual (which is alluded to in an exercise on Maxwell's equations), but the book stays pretty strictly away from the metric tensor and anything else remotely Riemannian, which I think is ultimately a good choice because it leaves room to discuss cohomology, Mayer-Vietoris, homotopy, etc.

This past year I took my first manifold theory/differential geometry course. We used John Lee's Introduction to Smooth Manifolds and the terse encyclopedic nature of the book didn't really help me understand what the professor was saying. Luckily, I found Loring Tu's book which gives a gentler introduction to the subject. Loring Tu's book has many computational examples and easy to medium level exercises, which are essential because of the onslaught of notation one encounters in manifold theory.I've been able to compare this book with John Lee's Introduction to Smooth Manifolds, which seems to be one of the standard texts for an introductory geometry course. My guess is that when Mr. Tu was writing his book, he started with John Lee's book and got rid of all of the obscure and difficult examples. He then expanded out the important essential ones in more detail so that a student who has never seen manifold theory would have a better chance of understanding.

This was a very accessible introduction to manifolds, but I felt that I could have been pushed more by the exercises. Any vaguely interesting or difficult exercise was accompanied with a hint that did all the hard work. If you don’t care about exercises, this book might be the easiest to read you can get.