Reviews (40)
A gentler introduction to smooth manifold theory than John Lee's
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.
Gentle introduction with good breadth and depth
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.
Great introductory book to start
I'm doing a phd in computer science and sometimes have a hard time reading math textbooks (I'm not bad at math just some math books need a completely different mindset) but this one is very well written. Each chapter is relatively short but contains all the essential topics.
Great Text and Clear Exposition
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.
Excellent introduction
This book is an excellent introduction to smooth manifolds. After reading this book and working through some of the exercises you will have a basic understanding of the language of smooth manifolds and be well prepared to delve into any number of topics including Riemannian geometry, Morse theory, symplectic geometry, contact geometry, Lie groups and algebras, and more advanced algebraic topology. Due to its clarity and the fact that it is fairly self-contained I found it well suited for self-study. In addition I quite appreciated how the book covers some algebra and provides definitions of things like algebras and modules. I also found the appendix on point-set topology to be quite useful.
Great !!!
It is remarkable! It is a complete book ! Let's get started.
Excellent.
This is an excellent book for what it is. A gentle yet rigorous introduction to the subject. I like the careful handling of various notion of orientation, the introduction to category theory which is unusual for a book of this level. An as others have pointed out, it is a good book for Bott and Tu's book on differential forms which is horrible in terms of introducing basic concepts. Some complain that the book is dry. It is somewhat dry, yes but that makes the book concise; think of it as learning the alphabet before you being to poetry. It is one of the best books in its category ;)
The new gold standard in theory of manifold
This is the best book of it's kind, It provides a solid introduction to manifolds.I used several books, Warner's "Foundations of Differentiable Manifolds and Lie Groups" and Jon Lee "Introduction to Smooth Manifolds" to name just few and I have to say that Tu's book is the best for several reasons: - He focus on the fundamental concepts of the theory and doesn't try to be encyclopaedic like Lee's book. - He introduce the calculus on manifold and Grassman Algebra throw Rn so every thing is clear and intuitive which is clearly not the case in Warner's book. - Most importantly the problems are designed to deepen one's knowledge of the theory and are designed carefully, for example some problems are guide the student to prove important theorems like the "Transversality theorem". If you want to understand what is a manifold don't buy anything else, just buy this one.
It's a very nice text. Exact and closed definitions
It's a very nice text. Exact and closed definitions, clear derivations of propositions and theorems. This work may be used as a textbook anybody who are interesting in different aspect of topology, abstract algebra and manifold.
A smooth introduction to manifolds
Manifolds are natural generalizations of smooth surfaces. Differential forms nicely summarize what kind of integrations are possible over a manifold. Stokes theorem is a beautiful generalization of classical theorems of vector analysis. In vector analysis, one meets the fact that whether a curl-free vector field has a potential or not in a specific domain depends on the topological properties of the domain (on simple-connectedness). This problem nicely generalizes to De Rham theory. Tu's book is a friendly and smooth introduction to these topics and more. I can recommend it to any student of mathematics who likes beautiful general mathematical concepts and has the patience and enthusiasm to understand a large number of definitions that this theory requires.
A gentler introduction to smooth manifold theory than John Lee's
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.
Gentle introduction with good breadth and depth
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.
