- Series: Universitext
- Paperback: 432 pages
- Publisher: Springer; 2nd ed. 2011 edition (October 6, 2010)
- Language: English
- ISBN-10: 1441973990
- ISBN-13: 978-1441973993
- Product Dimensions: 6.1 x 1 x 9.2 inches
- Shipping Weight: 1.7 pounds (View shipping rates and policies)
- Average Customer Review: 23 customer reviews
- Amazon Best Sellers Rank: #149,612 in Books (See Top 100 in Books)
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An Introduction to Manifolds: Second Edition (Universitext) 2nd ed. 2011 Edition
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From the reviews of the second edition:
“This book could be called a prequel to the book ‘Differential forms in algebraic topology’ by R. Bott and the author. Assuming only basic background in analysis and algebra, the book offers a rather gentle introduction to smooth manifolds and differential forms offering the necessary background to understand and compute deRham cohomology. … The text also contains many exercises … for the ambitious reader.” (A. Cap, Monatshefte für Mathematik, Vol. 161 (3), October, 2010)
From the Back Cover
Manifolds, the higher-dimensional analogues 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 second edition contains fifty pages of new material. Many passages have been rewritten, proofs simplified, and new examples and exercises added. This work may be used as a textbook for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. The requisite point-set topology is included in an appendix of twenty-five pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. Requiring only minimal undergraduate prerequisites, "An Introduction to Manifolds" is also an excellent foundation for the author's publication with Raoul Bott, "Differential Forms in Algebraic Topology."
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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.
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.
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 ;)
- 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.
so im having a good time with it.