Next semester (fall 2021) I am planning on taking a grad-student level differential topology course but I have never studied differential geometry which is a pre-requisite for the course. My plan is to self-study in the summer and this semester so that I do not have to waste a semester taking a differential geometry course which will ruin my schedule.
I am looking for a book that covers all of the following topics (ideally) but at least most of them:
any suggestions would be welcome.
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I had a good experience with self study as an undergrad using “Differential Forms and Connections” by Darling. It covers much of what you mention, and does a good job building from curves and surfaces to smooth manifolds, vector bundles etc. I agree with the comments that some of the topics listed in your quote are more in the direction of geometry than topology, and wouldn’t be necessary in a course following, say, Guillemin/Pollack, Tu’s “Introduction to Manifolds” or Hirsch’s “Differential Topology”.
Make sure your vector calculus is fully mastered, and your linear algebra, at least at the level of acing an intro course in each. Solid foundations make all the difference.
My experience with Michael Spivak's Differential Geometry was unpleasant and required a lot of use of his Calculus on Manifolds, which some love but I strongly do not recommend.
A big book meant for physicists is Misner Thorne and Wheeler's Gravitation, which I think is decent for intuition building. Unfortunately I haven't found a book on Differential Geometry that I actively like; MTW comes closest.