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.