An Introduction to Manifolds (Universitext)

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.

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.

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.

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 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.

It is remarkable! It is a complete book ! Let's get started.

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 ;)

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, 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.

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.

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.

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.

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.

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 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.

It is remarkable! It is a complete book ! Let's get started.

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 ;)

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, 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.

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.

Like other books on manifold, this book basically covers the same materials. But the good thing about this book is that, in chapter 1, it starts out with tangent space, cotangent space on R^n, which I think give most beginners a good intuition about the subject while most of the book jumps right into abstract manifolds.

After the painful "trench warfare" of reading Spivak's "Calculus on Manifolds" and still feeling like I still didn't fully grasp some of the more subtle details, Tu's book is a paragon of clarity in explaining the difficult concepts and constructions that form the foundations of modern differential geometry. While Munkres's "Analysis on Manifolds" is a great elementary exposition, suitable for students with limited exposure to higher mathematics and an excellent supplement/replacement for Spivak, this text by Loring Tu is a true introduction, making no demands on any previous knowledge of differential geometry, yet does not shy away from modern, fully general definitions, or the tools of analysis, modern algebra, and topology available to the contemporary mathematician. This text is aimed at advanced undergraduates and beginning graduate students with one year of real analysis and one term each of abstract algebra and topology as background, but its main purpose is the get beginning math grad students up to speed with the language of manifold theory. Thus, unlike some books, which purport only a year of calculus and a term of linear algebra as prerequisites, Tu assumes a certain degree of mathematical knowledge and sophistication on the part of his audience and does not sacrifice generality or clarity for the sake of maintaining an "elementary" presentation (Spivak's "Calculus on Manifolds" is a serious offender; even a very sophisticated book like Loomis and Sternberg's "Advanced Calculus" occasionally does this). This text does not seek to euphemize or hide any relevant concepts or definitions, but instead, presents them the way serious mathematicians work with them. Tu isn't afraid to introduce category theory, for instance, which works behind the scenes of and adds clarity to a large proportion of modern mathematics. Moreover, manifolds are defined in the context of Hausdorff spaces rather than as subsets of R^n. The sections on de Rham cohomology are especially excellent. Tu considers de Rham cohomology to be an especially important construction at the crossroads of algebra, geometry, topology, and analysis, and one of his goals was to ensure that students become comfortable with calculating cohomologies of reasonably "easy" spaces after finishing this book. Overall, this book gets students familiar enough with the basic strands of thought of smooth manifolds theory to comfortably pursue the rich literature of graduate textbooks afterwards, bridging the gap between undergraduate and graduate-level texts and courses. Two complaints not warranting the loss of a star. A) The exercises are a bit too routine, and it would be nice if a few very difficult problems (even if somewhat out of reach of a good student) were included. B) As a really trivial complaint, for some reason, the typesetting got worse from the first edition to the second, with poorer spacing and uglier Greek letters and special symbols.

I am what one might call a 'hobby mathematician' only, so please view my comments with this caveat in mind. I have tried several textbooks in an attempt to understand 'manifolds' - but I believe only this book managed to give me the understanding. The prerequisites are not very high, I think any advanced undergraduate math student will have them. 'Maps' play a major role in Prof. Tu's explanations, and they are always specified exactly and in such a way that one can understand their intuitive meaning, too. The proofs are detailed enough so one can follow them without many side calculations. It seems that if in doubt Prof. Tu added a few explanatory lines in his proofs, to make them easier to understand. The only subject I struggled with a bit was the last chapter 'de Rham cohomology' - this was not the book's fault, but it reflected the fact that I have never heared of this belfore. All in all, this is an EXCELLENT introduction to the subject of manifolds!

I borrowed this book from a library to learn differential geometry. So I cannot comment about Amazon's deliverance. As a Physics PhD student I should say that this book can be very helpful as long as one is aware that the purpose of the author is to teach differential geometry on a fast track. This means that some of the details of the each topic have not been covered. For instance the details of some of the proofs have been left as the exercises of chapter. However I don't think that this has damaged the main goal because the presentation of the material is still smooth enough such that one will have an acceptable level of understanding of differential geometry after finishing this book such that he/she would be prepared to learn some of the more advanced topics of this field after this book. This was exactly my purpose from reading this book which is completely satisfied and that's why I am giving it 5 stars. However if you have more time and are eager to learn all the details, the book by John Lee "Introduction to Smooth Manifolds" , would serve you better because he covers almost the same material (well, there are three chapters which have not been mentioned here at all) but in over 600 pages. Jeffrey Lee's book, "Manifolds and Differential Geometry" is also a nice book esp someone wants to learn Riemannian geometry too. About the prerequisites, while I had some background in point set topology from Munkres, but it was not really necessary because there is a short appendix at the end which has reviewed the required concepts of topology. A solid background in Algebra and Analysis would be necessary though. Overall, reading this book is relatively easily and is very well suited for self-study. The exercises of each chapter are relatively easy and sometimes introduce some new concepts.

I have read a lot of books about differentiable manifolds. I only found this book giving most detailed and down-to-earth approach. I think every graduate students in Mahs should read and acquire knowledge from this book no matter what area researches you do in Pure Maths. I hope Tu can write more books in this style. It will be a good news to every Maths students if he can write a introductory text on algebraic geometry.

Let me compare it to other textbooks on manifold theory - Compared to John M. Lee's "Introduction to smooth manifolds", Loring W. Tu covers pretty much the same amount of material in merely 300 pages as does Lee in 600+ pages. Lee tries to insert motivational stuff and intuitive explanations, but oftentimes they become to verbose and (at least for me) confusing. Tu manages to keep exposition as clear as possible without becoming to terse. Compared to A. Wallace's "Differential topology. First steps", Tu's book uses modern tools like a bit of category theory and, anyways, it is more detailed. Compared Guillemin and Polack's "Differential topolgy" and Milnor's "Topology from differentiable viewpoint" Loring Tu covers a bit different material, which in my view is more appropriate for a student learning manifold theory for the fist time: i think topics like transversality and intersection theory are better digested AFTER one has read Tu's book. To sum up, Loring Tu has managed to compile a very balanced text: in only 300 pages he covers basics of manifold theory, in a very clean modern way using excellent notation (that is very important, because many textbooks on manifolds and differential topology use horrible confusing notation). The text can be read right after a good course in analysis (like Rudin) and linear algebra (like Axler's "Linear algebra done Right"). The only drawback is, in my opinion, lack of challenging exercises.

The author seems to do everything right. He works in Euclidean spaces then progresses to manifolds. This books filled in so many gaps for me. If you are prepared to take this book chapter by chapter, the rewards will be incredible.

This book is very detailed while concise. It is a better read than John M Lee smooth manifolds for beginners.

Very good and easy to get idea.

I can confirm that this text holds up as a first foray into manifolds, a parallel text alongside a course, and a reference once the course is over. Few math texts can boast that. It accomplishes this while being refreshingly economical in its writing. For supplementary reading, you may want to consult Lee's book on Smooth Manifolds, although his treatment can be a bit *too* chatty. Note that the treatment of tensors, tangent vectors and differentials may differ from your professor's favorite approach. This is not a smear on the Tu's text, but rather a by-product of the many (equivalent) approaches to such concepts. EDIT IN 2020 Tu's book is basically a rewrite of Boothby's Introduction to Smooth Manifolds and Riemannian Geometry. I would give that book five stars and this one I have downgraded to four. Boothby is the "go-to" text for good reason: it is clearer and furnishes more examples than Tu's.

Very well written book, I can recommend this product to anybody starting a postgraduate degree in anything STEM related, or even as a book to pick up if you're interested!

Forty Years ago, such an Introduction to Differential Geometry did not exist. The Historical context of the subject had to be mined out of hard rock. Despite doing some original research at the time, there was always a feeling that I was just scratching the surface. Pun intended. Today there are Introductory Texts that quickly orientate the student to the Historical Background, and brings insights quickly to the surface. TU's Introduction to Manifolds must be the clearest of modern texts designed for Graduate Students. For undergraduates, it is probably too condensed to serve as a first Text on the subject. It is highly recommended.

Un libro muy bueno. Me fue de mucho provecho desde los primeros capítulos, en los cuales habla acerca del álgebra tensorial necesaria para entender las variedades. Haciendo una explicación más detallada y moderna que otros autores. Por otra parte, me gustó el hecho que tenga incluído un apéndice sobre topología, el cual me fue de mucha utilidad para entender mejor ciertos conceptos a lo largo del libro. La única razón por la que no le doy 5 estrellas, es porque esperaba que la calidad de impresión sea un poco mejor. Aunque no es gran cosa en realidad.

Consiglio questo testo a tutti gli studenti italiani dei corsi di Matematica e Fisica. Soprattutto a questi ultimi, se ritengono che un testo che, pur mantenedo il rigore di una trattazione matematica non perde mai di vista il senso storico e logico nello sviluppo dei contenuti, possa costituire un "viatico" per accedere ai piani "alti" del corso di studi che hanno scelto. Rigoroso, accessibile, self-contained. Adatto anche ad un progetto di autoformazione

If you want to know many basic topics in differentiable manifolds, then this is your book. And contains also De Rham Cohomology.

Definitely the best text of manifolds for an undergraduate. Also good for a graduate student who needs an easier and more slow-paced companion to Lee's book on smooth manifolds.

Très bon livre d’introduction ! Achat pas parfait !

Softcover book

この本は微分幾何学を学ぶすべての人におすすめです。 Chap１では微積分、線形代数や一般位相についてまとめているため基本事項の復習もばっちりです。また、Chap２以降では多様体の一般論について多くの具体例を紹介しながら細かく解説してあります。私も多様体について勉強していたときは、この本の具体例に沿って手を動かしながら理解を深めていきました。多様体について勉強をはじめる学部３年生や、専門分野の中で多様体の知識が必要になった学部４年生もぜひこの本を手に取ってみてはいかがでしょうか。

　Tu の入門書は、彼自身が冒頭で述べているように、Bott との共著『Differential Forms in Algebraic Topology』の第1章への導入を意識した内容となっています。第7章のド・ラーム理論のみならず、随所にBott との共著の第1章を理解するのに役立つ内容が並んでいます。また、全体の構成をみても、非常に洗練された現代的な内容となっています。 　しかし、当該本は、説明項目は入門書のレベルを超えて多岐にわたっていますが、個々の説明は簡潔かつ洗練されているが故に、初学者のレベルでは、その本質的部分を見過ごすことが起きる可能性を秘めています。実際、小生は当該本を勉強した上で、次に松本を手に取りましたが、特に写像の微分について、松本を読んで初めてその本質に気づきました。松本の入門書の実直な書き方はくどいとの指摘もありますが、多様体の基本的な項目について丁寧に掘り下げた説明をしているという点で、初学者にとって誠に有難い存在です。 　小生は、このTuの入門書の日本語版について書かれた大類氏のレビユーに共感する所も多く、初学者は、まず松本で本質的な部分を押さえた方が真に理解が進むような気がします。　 　ただし、Tuの入門書は、より高いレベルに進むための、最高の現代的な案内役であることには間違いありません。例えば、当該本の第7章では、Mayer-Vietoris 系列について、Bottとの共著の第1章の説明を補完するような門外漢にも理解できるような丁寧な解説を試みています。