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Curvature in Mathematics and Physics (Dover Books on Mathematics) Paperback – August 22, 2012

4.4 out of 5 stars 17 customer reviews

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Editorial Reviews

About the Author

Sternberg is George Putnam Professor of Pure and Applied Mathematics, Harvard University and Permanent Sackler Fellow University of Tel Aviv.
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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 416 pages
  • Publisher: Dover Publications (September 19, 2012)
  • Language: English
  • ISBN-10: 0486478556
  • ISBN-13: 978-0486478555
  • Product Dimensions: 0.8 x 6.2 x 9.2 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (17 customer reviews)
  • Amazon Best Sellers Rank: #328,256 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

By Let's Compare Options Preptorial TOP 500 REVIEWER on February 1, 2013
Format: Paperback Verified Purchase
A few reviews here and around the web characterize this as "undergrad." At MIT? If you don't have STRONG linear algebra and advanced (minimum 3 years) calculus, you'll be lost by the third chapter.

I'm the CTO of Classpros dot com and design math visualizations for Engineering students and this is NOT an undergrad text, unless you're an EXCEPTIONAL undergrad with a LOT of physics and calc already behind you, as in a senior at a tech college. This is not at all to knock the book, it's to save purchasers from disappointment when they see the author jumping into Relativity from the viewpoint of Cartan exterior calculus and tensors.

That said, this is an EXTRAORDINARY text because differential geometry has become so specialized that few grad students except in limited areas of physics/applied math get to go there. The "expansion" of the field with game programming and sims is a new revolution that is "bringing back" fields as dusty as quaternions and spherical trig (see our article on Wiki on the Lenart Sphere, for example)-- and creating exciting new interest in differential forms and geometry.

After reviewing and using half a dozen (rare and old) books on differential geometry, this is the ONE book you must have if you are serious about the field, both for breadth and depth and especially currency. Both the most recent applications and the older physics are covered flawlessly. Also, if this were a Springer text, it would be over $100 at this quality and rarity-- what a bargain from Dover!

Several reviewers have noted this text as worth the price even though it is a "classic." Ahem.
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Format: Paperback
The 'search inside' feature is missing the table of contents.

Here it is:

Introduction
1. Gauss's Theorem Egregium
2. Rules of Calculus
3. Connections on the Tangent Bundle
4. Levi-Civita's Theorem
5. Bi-invariant Metrics on a Lie Group
6. Cartan Calculations
7. Gauss's Lemma
8. Variational Formulas
9. The Hopf-Rinow Theorem
10. Curvature, Distance and Volume
11.Review of Special Relativity
12. The Star Operator and Electromagnetism
13. Preliminaries to the Einstein Equation
14. Die Grundlagen der Physik
15. The Frobenius Theorem
16. Connections on Principal Bundles
17. Reduction of Principal Bundles
18. Superconnections
19. Semi-Riemannian Submersions
Bibliography
Index
1 Comment 108 people found this helpful. Was this review helpful to you? Yes No Sending feedback...
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Format: Paperback
While one cannot exactly call this an easy introduction to differential geometry, its many problems and proofs cover such a wide range of topics in the field that when the reader successfully completes it, there is no longer much need for an introductory text -- except as reference.

Far more important for me was brushing-up on (and then tightening-up on) both linear algebra and Group Theory, -- at least up through Lie Groups. Breathing space for this was more or less allowed in the problems section at the end of chapter 2. Without this warm-up session, trying to apply Riemann geometry to a Euclidian space in chapter 5 would have amounted to little more than a "distant abstraction."

The challenge throughout the book was making the connection in ones mind of the dizzying array of curvatures associated with hyper-surfaces possible to embed in Euclidian space. I finally figured out that doing this through the "Calculus of Manifolds" is what this book was all about, and thus represented the true value of the book?

In any case, whether true or not, I was delighted to see both the "Lie Derivation of Vector Fields," and the "Schwarzschild Solution" worked through. Previously I had never seen a proper analysis of the former, and the latter only as "settled formula" -- without ever having seen its derivation.

So far, I have completed the book up through chapter 12, more enlightened than frightened.

When the masters of a subject (such as Shlomo Sternberg is here) presents his material, there is the natural expectation that the reader is going to have to stretch a bit to keep up.
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Classic book that will provide you with a rigorous overview of the links between the conceptual properties and interesting empirical applications that make the study of the physics and mathematics of nonlinear surfaces really exciting. I think it's one of the best -- if not the best -- book of its kind. It was a brilliant move to include this outstanding book in the Dover series makes because that increases accessibility since the price of each book is affordable unlike most of the comparable excellent titles at say Springer. I've gained the ability to start applying this work to biological problems in microbiology (bacterial cell membrane properties such as protein permeability). I highly recommend this book.
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Yes, it is a superb explanation of the subject I am investigating on.; of course it is extremely technical, no room for beginners like me but still, working hard I am getting a good satisfaction out of it!
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