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Smooth Manifolds and Observables (Graduate Texts in Mathematics) 2003rd Edition

5 out of 5 stars 1 customer review
ISBN-13: 978-0387955438
ISBN-10: 0387955437
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Editorial Reviews

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"Main themes of the book are manifolds, fibre bundles and differential operators acting on sections of vector bundles. … A classical treatment of these topics starts with a coordinate description of a manifold M … . The present book is based on an alternative point of view, where calculus on manifolds is treated as a part of commutative algebra. … The book contains quite a few exercises and many useful illustrations." (EMS, September, 2004)

"The book provides a self-contained introduction to the theory of smooth manifolds and fibre bundles, oriented towards graduate students in mathematics and physics. The approach followed here, however, substantially differs from most textbooks on manifold theory. … This book is certainly quite interesting and may appeal even to people who merely want to study algebraic geometry, in the sense that they will gain extra insight here by the attention which is paid to making certain constructions in algebraic geometry physically or intuitively acceptable." (Willy Sarlet, Zentralblatt MATH, Vol. 1021, 2003)

Language Notes

Text: English (translation)
Original Language: Russian
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Product Details

  • Series: Graduate Texts in Mathematics (Book 220)
  • Hardcover: 222 pages
  • Publisher: Springer; 2003 edition (October 4, 2002)
  • Language: English
  • ISBN-10: 0387955437
  • ISBN-13: 978-0387955438
  • Product Dimensions: 6.1 x 0.6 x 9.2 inches
  • Shipping Weight: 15.2 ounces (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Best Sellers Rank: #2,903,503 in Books (See Top 100 in Books)

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Format: Hardcover
This text serves as an introduction to the theory of smooth manifolds, but it's quite unlike any other text on the subject. The classic approach to manifold theory is heavy on the (point-set) topology and analysis; this one is heavy on the algebra. In this book, the author shows that a manifold (in the traditional sense) is completely characterized by its ring of smooth functions. The question then becomes: which rings can be constructed in this way? The author answers this question and, in the process, ends up building a completely algebraic theory of manifolds--it's totally awesome.
What's more, he uses the idea of physical observables to motivate the construction. This is a delightful surprise, since algebraists don't tend to focus much on applications of a theory. Indeed, the concept of physical observation is brought up in the first chapter and used to justify many of the algebraic constructions throughout the book. A traditional manifold book would probably leave the entire discussion of mechanics (and more generally, symplectic structures)until the end (if it's included at all).
Though the technical prerequisites are modest, I suggest you study some other texts first (or concurrently). The traditional approach to manifolds is still quite important and (as of yet) irreplaceable. I suggest you study this either before or along with this text. Indeed, you might get more out of this book if you are already familiar with the traditional theory. I suggest John Lee's "An Introduction to Smooth Manifolds"--the best out there in my opinion. Furthermore, you should make sure your algebra is sharp before you undertake this book. A graduate level understanding should suffice (something on the order of Dummit/Foote's Abstract Algebra text should do quite nicely).
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Smooth Manifolds and Observables (Graduate Texts in Mathematics)
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