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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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    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.2012-07-13

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

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    @Kyle A book that covers similar material but is a little less condensed in Analysis on Manifolds by Munkres2012-07-13
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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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    Just for comparison, Spivak does not mention vector bundles, submanifolds, Lie groups -- and his definition of "manifold" is "an embedded submanifold of $\mathbb{R}^n$."2012-08-01
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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.