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I learn mathematics by myself, and had recently been using do Carmo's "Differential Geometry of Curves and Surfaces". I had to give up the book, I found it disorganized and difficult to understand. I have picked up O'Neil's book and so far, despite the completely different notation, I've been finding it alright, and I wonder if it was the correct choice.
I already have some background on analysis, calculus, metric spaces and linear algebra, and I wish to learn the modern differential geometry (as in, the stuff after Gauss), particularly stuff about covariants, tensors, connections and etc. Am I "jumping a step" by not learning classical differential geometry?

J. W. Tanner
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khalelbm
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    I personally think that you will understand graduate level differential geometry material (abstract manifolds, connections on vector bundles, etc.) much better if you have mastered the basic undergraduate material. That's where the intuition gets developed. Allow me to suggest you look at my text, freely available at the link in my profile. O'Neil does everything with differential forms and Cartan's structure equations, which are very much to my taste; but I don't recall his having particularly interesting exercises. My text does :) – Ted Shifrin Jan 07 '25 at 04:32
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    Let me second Ted's suggestion, even though I've never seen his book. Even back in grad school where we met many years ago, he had a talent for interesting questions and clear exposition. I imagine he's put that talent to good use over the years. :) – John Hughes Jan 07 '25 at 11:25

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I often recommend O'Neil as a good first place to read stuff. I like Millman and Parker as a second book --- everything you'll read, you'll recognize as "O'Neil with more generality". Both are at their best if you do most of the exercises.

At some point, anyone learning differential geometry should probably read the exposition in Milnor's Morse Theory (at least I think it's in there). You'll recognize things there as "just what I already learned, but without any indices on anything!", which is kind of a revelation. (Also kind of a curse: if you learn the material only this way, you may find yourself wondering "OK, now how would I compute the Gaussian curvature at a point of the torus in 3-space?" and realize you had no idea.)

John Hughes
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