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I'm entering my 5th undergrad semester and regarding topics somewhat related to differential geometry I so far have heard calc I-III (measure theory etc.), introduction to PDEs (harmonic function theory, heat equation, ...), complex analysis and introduction to topology and geometry (set-point, algebraic topology up to the classification of path-connected coverings and deck transformations, geometry up to 2nd fundamental form and the theorema egregium). Next semester I will take a course called "introduction to analysis and geometry on manifolds". The literature mentioned as part of the course description goes as follows:

  • R. Abraham, J. Marsden, T. Ratiu: manifolds, tensor analysis, and applications (Springer 1998)
  • G. Bredon: topology and geometry (Springer 1997)
  • J. Lee: introduction to smooth manifolds (Springer 2003)

I want to use my 2ish months of summer break to prepare for the course and dive deep into differential geometry, planning to maybe even write my thesis in the field. I have a pretty poor algebraic background, not going much beyond linear algebra II at all. I still find a lot of the algebraic concepts of my topology course very fascinating, I just know I'll not be able to comprehend deeper theory (the amalgamated product in SvK2 still leaves me clueless for reference) until I've done some brushing up on the side. Regardless, I love the geometric ideas and hints of algebra and am not so much interested in super technical, calculation/analysis heavy literature as I am used to from PDEs for example.

Which of the above books could be fitting for me? Do you have any other suggestions? I'd like it to prepare me for said course but also give me a good general overview, not minding "splashes" of algebra, more so than it being 100% analytical. Any help appreciated.

Suzu Hirose
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Gérard
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    https://math.stackexchange.com/q/46482/305862 – Jean Marie Jul 25 '22 at 20:39
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    You will need to get comfortable with tensor products (of vector spaces, at least) and exterior algebra. You can't avoid tensors and differential forms in differential geometry. I think Bredon includes a lot of the material in Guillemin and Pollack's Differential Topology, but start there before doing full-blown abstract smooth manifolds. And the degree/intersection theory material is powerful and beautiful. – Ted Shifrin Jul 25 '22 at 21:09
  • I second @TedShifrin's recommendation of Guillemin-Pollack (but beware of the errors and gaps). – Deane Jul 25 '22 at 21:48
  • Re errata, a list is available on my website, listed in my profile. I am not sure if @Deane has a list of the gaps I don’t know. :) I taught out of the book numerous times, but presented my own proofs of some theorems (like Poincaré-Hopf). – Ted Shifrin Jul 25 '22 at 21:49
  • @TedShifrin, you have an errata for GP?? How long have you had this available? I never realized there were any serious gaps until I had a very good undergraduate student study it. I was amazed by how many issues he found. I wish I had known about your errata. – Deane Jul 25 '22 at 21:52
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    @Deane Decades. I tried to get Victor to correct things after a few years, but he never did. Or P-H dropped the ball. If you have more for me to add, please email them to me! – Ted Shifrin Jul 25 '22 at 21:53
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    Consider Noel J. Hicks' notes https://www.wisdom.weizmann.ac.il/~yakov/scanlib/hicks.pdf – janmarqz Jul 26 '22 at 18:28

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