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An Introduction to Manifolds (Universitext) Paperback – 29 Oct 2007

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Product details

  • Paperback: 384 pages
  • Publisher: Springer (29 Oct. 2007)
  • Language: English
  • ISBN-10: 0387480986
  • ISBN-13: 978-0387480985
  • Product Dimensions: 15.5 x 2.2 x 23.5 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Bestsellers Rank: 1,683,219 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Product Description


From the reviews: "An introduction to the formalism of differential and integral calculus on smooth manifolds. … Many prospective readers of Bott and Tu will welcome this volume. … Summing Up: Recommended. Lower-division undergraduates." (D. V. Feldman, CHOICE, Vol. 45 (10), June, 2008) "An Introduction to Manifolds is split up into eight parts, well organized, well written, and, as Tu claims, readable. … This excellent and accessible book also comes equipped with plenty of examples and exercises, whence it will serve well as both a classroom text and a source for self-study. Indeed, I propose to use it myself, given that I am one of the non-experts … ." (Michael Berg, MathDL, April, 2008) "A book which … covers all the essential topics in differentiable manifolds theory, and sufficiently elementary so that it can be read and understood with only minimal prerequisites—all this in less than 360 pages. The book is divided into seven parts, plus four appendices. … The added value of the book lies mainly in the simplicity, the clearness and the concision of the exposition. … is certainly one of the most readable introductions to differential geometry." (Ahmad El Soufi, Mathematical Reviews, Issue 2008 k) "The textbook under review is very well-written and self contained. … It extends the calculus of curves and surfaces to higher dimensions. The higher dimensional analogues of smooth curves and surfaces are called manifolds. … 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." (Ion Mihai, Zentralblatt MATH, Vol. 1144, 2008)

From the Back Cover

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, An Introduction to Manifolds is also an excellent foundation for Springer GTM 82, Differential Forms in Algebraic Topology.  

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Most Helpful Customer Reviews

4 of 4 people found the following review helpful By I. D. PLATIS on 11 Mar. 2010
Format: Paperback
L.W. Tu's book is probably the best today's book that one can find among the introductory books in Differentiable Manifolds. Thoroughly written in a modern and comprehensible way, with several examples and useful appendices, leads the reader (and the teacher) straight to the very essence of the object. I recommend it with absolutely no hesitation.
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0 of 1 people found the following review helpful By Philip on 12 Nov. 2014
Format: Paperback Verified Purchase
It seems a bit less terse then Lee which makes the going a bit easier. (nothing against Lee's book, just good to have an alternative excellent treatment)
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Most Helpful Customer Reviews on (beta) 15 reviews
48 of 49 people found the following review helpful
Clear and Solid Exposition 22 July 2008
By pnmti - Published on
Format: Paperback
This is my favorite book on Differentiable Manifolds. After reading this book the reader will obtain a solid background on the following essential notions: Charts and atlas of a manifold; tangent vectors (as derivations); differential of a smooth function between manifolds; submanifolds and embeddings; quotient spaces; partitions of unity; vector fields; vector bundles; differential forms and de Rham cohomology. And on the road, the reader gets a gentle exposure to Lie groups, Lie algebras; and some basic notion of Category and Functors.

I found the following aspects of the book especially attractive:
> Clear style of writing: The author is the coauthor of the acclaimed "Differential Forms in Algebraic Topology". See the comments for that book. The clarity has not decreased at all.
> Bite-sized sections: The materials contained in each section is approximately equal to that of a 50-minute lecture. This helps readers who plan self-study.
> Right amount of topics: This is not an encyclopedia on manifolds. However, it does contain the ``absolute must'' one should know about manifolds. And it does such a good job in presenting it, the reader will be left with a solid understanding on those essential topics.

I first read this book as a Physics student and had no trouble reading it. I later switched discipline to Mathematics, and I know that this book has helped me appreciate the beauty of Mathematics. I thank the author for writing such an wonderful book.
29 of 30 people found the following review helpful
fills a gaping hole 14 April 2008
By chicken head cut off - Published on
Format: Paperback Verified Purchase
i think there is a jump from ugrad analysis/alg/top etc to early grad school concepts. i didnt know category theory, i only had the flimsiest notion of a manifold, etc etc. and this book fills in that jump wonderfully. it does the right mix of analysis-differential topology-topology so that you can go read a book like bott and tu later (that's what it was designed for).
so im having a good time with it.
31 of 34 people found the following review helpful
An introduction it Manifolds 1 May 2008
By Gregory S. Chirikjian - Published on
Format: Paperback
This is an excellent book. I wish that more books on advanced mathematics were written in this style. In contrast to most books on manifolds that tend to be very difficult for beginners to follow, Prof. Tu has made every effort to make this subject understandable to the nonexpert.

Greg Chirikjian
Professor, Mechanical Engineering
Johns Hopkins University
12 of 12 people found the following review helpful
A smooth introduction to manifolds 14 Mar. 2011
By Tamas from Hungary - Published on
Format: Paperback Verified Purchase
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.
11 of 11 people found the following review helpful
Great Text and Clear Exposition 3 Feb. 2013
By CJ Lungstrum - Published on
Format: Paperback Verified Purchase
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.
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