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I'm going to be taking a graduate course in differential geometry, this coming fall, but I am not prepared for it. Can anyone recommend a good introductory treatment of the background materials?

The list of topics in the course is:

Manifolds, Local Study of Manifolds, Vector bundles, Submanifolds, Vector Fields, Lie Groups (brief treatment), Differential forms, Orientation and Integration, Statement of the Hodge Theorem, The Kähler condition

My calculus background (particularly advanced calculus) is not strong. I had a three semester coverage of calculus (the typical Calc 1, Calc 2, and Calc 3) and this was almost a decade ago. Since then my experience has been almost exclusively with pure math -- applied math courses always made me uncomfortable.

I have about a month to prepare for this course so I'd like to make as much of this time as possible; and arbitrarily choosing books on the topic is a great way to waste time I've found.

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What do you think will be "applied" in this course? I don't see any General Relativity in your topic list. I think topology and algebra are fine prerequisistes here. Note that the differentials and integrals you will encounter here are abstract operators that you should have plenty of experience with from an (abstract) algebra course. They are used to establish local-to-global relationships, which appear to be the point of the last part of the class. –  ex0du5 Jul 13 '12 at 1:11
I sometimes (probably in error) include calculus with the phrase "applied math". I hope that my error is not taken as a slight against anyone's chosen field of expertise. My remark about applied math was only to give a clearer picture of my background. –  roo Jul 13 '12 at 1:16
Make sure that you are thoroughly up-to-date in linear algebra. In particular, make sure you have a full understanding of duality, inner product spaces and tensor/exterior products, up to the level covered in, say, Rotman's Advanced Modern Algebra or Birhhoff/Mac Lane's Algebra. –  ItsNotObvious Jul 13 '12 at 11:49

3 Answers 3

up vote 6 down vote accepted

Spivak's Calculus on Manifolds is a great little book.

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I skimmed this book's table of contents and read the introduction and it sounds perfect for my needs. Thank you! –  roo Jul 13 '12 at 1:13
I found it enjoyable and productive. Buono Appetito! –  ncmathsadist Jul 13 '12 at 1:26
@Kyle A book that covers similar material but is a little less condensed in Analysis on Manifolds by Munkres –  ItsNotObvious Jul 13 '12 at 11:52

I think you should revise multivariable calculus subjects.Vector Calculus, Marsden and Tromba might be helpfull for you.If you just want an insight without rigorous proofs Penrose's the road to the reality is really wonderfull source. Finally, you can read John M, Lee, introduction to smooth manifolds, it is an introductory book on this subject.

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+1 My recommendation exactly: Marsden-Tromba is very good for multivariable calculus, and Lee's "Introduction to Smooth Manifolds" is excellent. Also, Lee's book covers way more of the topics you mentioned than Spivak's, and has an appendix in the back with all of the necessary pre-requisites. –  Jesse Madnick Aug 1 '12 at 1:19
Just for comparison, Spivak does not mention vector bundles, submanifolds, Lie groups -- and his definition of "manifold" is "an embedded submanifold of $\mathbb{R}^n$." –  Jesse Madnick Aug 1 '12 at 1:19

I asked a question on http://math.stackexchange.com approximately one year ago in which I requested for suggestions for good (theoretical) multivariable calculus textbooks. The link to the question is:

(Theoretical) Multivariable Calculus Textbooks

You might find the answers to the question interesting.

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