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I was wondering if anyone could recommend some books for studying topics such as abstract manifolds, differential forms on manifolds, integration of differential forms, Stokes' theorem, de Rham cohomology, Hodge star operator? Our text is A Comprehensive Introduction to Differential Geometry by Spivak, but I think this book is very difficult for a beginner to learn.

Thanks in advance.

Martin Sleziak
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user53109
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  • You might want to look at: http://math.stackexchange.com/questions/13575/teaching-myself-differential-topology-and-differential-geometry and http://math.stackexchange.com/questions/17969/differential-geometry-of-curves-and-surfaces-bibliography. Regards – Amzoti Dec 24 '12 at 00:33
  • Teron Hitchman has been reviewing a fairly decent list of differential geometry textbooks on his blog. Highly recommended: http://theronhitchman.wordpress.com/ – Andrés E. Caicedo Dec 24 '12 at 01:27
  • Not really a differential geometry book, but Munkres' "Analysis on Manifolds" is very elementary and well explained, and covers all the topics you listed except the Hodge star operator and may not go as deeply into de Rham cohomology as you might like. It could be a good introductory book that give you enough of a running start to use your course text. – Ragib Zaman Dec 24 '12 at 01:55
  • Perhaps some other questions tagged differential-geometry+reference-request might be of interest for you, too. – Martin Sleziak Dec 24 '12 at 11:22

4 Answers4

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My favourite book on the subject is Introduction to Smooth Manifolds by John M. Lee. It is quite explicit, which is sometimes what you need when you first start to learn a topic, as you haven't developed any intuition yet. Also, Lee's writing style is excellent, in particular, he does a great job of motivating each topic. I like this book so much, my parents are buying me a copy for Christmas.

Added later: I just checked, and it seems that the book doesn't have that much about the Hodge star (only exercises 12-16 in Chapter 14). I'm not sure how much you need to know about the Hodge star, but the exercises cover the standard facts.

Michael Albanese
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    I used this book when I took a class in geometry and liked it a lot. Highly recommended. – Potato Dec 24 '12 at 01:31
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    This is indeed an excellent textbook, I learnt most of the differential geometry I know from it. – Olivier Bégassat Dec 24 '12 at 01:47
  • Thanks again Michael! I don't know how much my professor will cover Hodge star. It's a brand new topic to me. So I will see how it goes and may have a better sense later on. – user53109 Dec 24 '12 at 15:26
  • As mentioned in another answer, one book with more detail on the Hodge star and theorem is Warner's book; it's difficult, but rewarding. –  Apr 11 '15 at 16:09
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There's no arguing that John Lee's texts are excellent: the following are part of the series "Graduate Texts in Mathematics":

Each of the above links to Amazon, simply because you can preview the texts, e.g., the Table(s) of Contents, to see if any/all meet your needs. Each is also accompanied by credible "reviews", which may help you select the appropriate text(s) to meet your needs.

As you seem to be looking for a more elementary introduction to differential geometry:

You might want to check out the the course on Differential Geometry via MIT Open Course Ware, (Prof. Paul Seidel):

This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.

Other choices for Elementary Introductions:

  • You might also want to look into O'Neill's Elementary Differential Geometry, perhaps a good choice to start off with.

    "Written primarily for students who have completed the standard first courses in calculus and linear algebra, it provides an introduction to the geometry of curves and surfaces."

  • Also look into the book with the same title: Elementary Differential Geometry, 2nd Ed (2010), [Springer Undergraduate Mathematics Series], this one authored by Andrew Pressley.

    "Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout."
amWhy
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  • Which one in your opinion would be better to start with: "Introduction to Smooth Manifolds" or "Manifolds and Differential Geometry" (both by M. Lee). – Learner Dec 24 '12 at 01:14
  • @Learner: Between the two, probably start with Intro to Smooth Manifolds... – amWhy Dec 24 '12 at 01:16
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    If you get Pressley's book, be absolutely sure to get the second edition. The print and typesetting quality of the first edition leaves a lot to be desired. – kahen Dec 24 '12 at 14:30
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    FYI: "Intro to Smooth Manifolds" and "Manifolds and Differential Geometry" are by DIFFERENT Lees :-) –  Aug 07 '14 at 23:57
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In that you mention "intro," here is a link to an excellent book on differential geometry (free download) about 1/3 down Prof. Shifrin's homepage. It is quite highly regarded and clearly written for self-study:

http://math.uga.edu/~shifrin/

Pete L. Clark
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A fine old book which hasn't been mentioned, which treats the topics asked for and more, is Frank Warner's Foundations of Differentiable Manifolds and Lie Groups. Maybe harder than Spivak, but Warner does a lot in a snappy way in not too many pages.

John
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