An Introduction to Manifolds (Universitext Book 0) 2nd Edition by Loring W. Tu (PDF)

Description

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory.In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems.This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, ‘Introduction to Manifolds’ is also an excellent foundation for Springer’s GTM 82, ‘Differential Forms in Algebraic Topology’.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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.

⭐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.

⭐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 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.

⭐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 2020Tu’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!

⭐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.Bought this book since my university completely didn’t care to teach it’s mathematicians any geometry beyond Euclidean… For me this book is quite concise I worked trough the entire book during the last two weeks. It consists of a lot of small subsegments that are easily understood. Not too much unnecessary text very well structured. Cannot say how understandable it is for non mathematicians, however it for for me self studying geometry. Will see how it works now as reference manual.

⭐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. Punintended.Today there are Introductory Texts that quickly orientate the student to the Historical Background, and bringsinsights 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.

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