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I am soon going to start learning differential geometry on my own (I'm trying to learn the math behind General Relativity before I take it next year). I got the sense that a good, standard 1st book on the subject was do Carmo's Differential Geometry of Curves and Surfaces and so that was the book I planned on reading. However I just read this question on mathoverflow, and both answers to it suggested that the professor NOT teach a class from a book like do Carmo's because it doesn't cover differential forms.

Would you guys agree that I should find a book that introduces differential forms (and tensors?) given that I am an undergrad physics major who plans to study relativity theory? If so, what books would you recommend?

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  • $\begingroup$ There's Geometry of Spacetime by I think Callahan. It's pretty good but not sure if it covers forms. Loring Tu and Lee both have books on smooth manifolds that definitely cover forms, and docarmo has a book on just forms $\endgroup$ – Ashley Jul 24 '14 at 22:29
  • $\begingroup$ Spivak's A Comprehensive Introduction to Differential Geometry is never a bad choice. $\endgroup$ – rogerl Jul 24 '14 at 22:33
  • $\begingroup$ You have to decide what type of differential geometry you are interested in. You can study "classical" differential geometry or "modern" differential geometry. A rule of thumb says that classical is things Gauss knew while modern is everything after Gauss. $\endgroup$ – Brad Jul 24 '14 at 22:50
  • $\begingroup$ @Brad I assume modern would be the better choice wouldn't it? Mathematicians know more now and can choose better methods -- I would guess. Plus modern papers are probably written in the language of "modern" differential geometry. Is there a reason why I'd want to study "classical" differential geometry given the choice (I really don't know that difference between the two so that's not a rhetorical question)? Are differential forms the "classical" or "modern" approach? $\endgroup$ – user166203 Jul 24 '14 at 22:51
  • $\begingroup$ @rogerl I just looked up Spivak's books and man! They have some AWESOME looking covers. I guess I really never got that "don't judge a book by its cover" thing because I can't imagine books that look that great could be anything but. ;) $\endgroup$ – user166203 Jul 24 '14 at 22:57
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I'm going to agree with Bryant in the mentioned link and recommend O'Neill's Elementary Differential Geometry. It is a gentle enough introduction to differential geometry, uses the common language and will prepare you for the usual problems in $\Bbb R^3$ while giving you a hint of what comes next.

It may be profitably followed by his second book and/or John Lee's Introduction to Smooth Manifolds and Riemannian Manifolds.

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In my opinion the best Differential geometry book is John M. Lee - Introduction to Smooth Manifolds followed by Loring W. Tu - Introduction to manifolds and Jeffrey M. Lee - Manifolds and Differential Geometry.

For connections and Riemannian Geometry look also John M. Lee - Riemannian Manifolds: An introduction to curvature.

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  • $\begingroup$ John Lee's book is in the GTM series. Is an undergrad physics major like me going to know enough math to handle that? For reference, I've taken the calculus sequence, ODEs, linear algebra, and an intro to real analysis but that's all the math I have to take. $\endgroup$ – user166203 Jul 24 '14 at 22:41
  • $\begingroup$ Maybe Tu's book is a better choice in this case. $\endgroup$ – Dubious Jul 24 '14 at 22:44
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    $\begingroup$ OK, thanks. I'll check it out. :) $\endgroup$ – user166203 Jul 24 '14 at 22:47
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    $\begingroup$ Although I'm pleased to see my books recommended, personally I wouldn't use them for an undergrad course in curves and surfaces. In fact, I wouldn't recommend using forms in that course at all, despite the advice from Bryant (who's one of the smartest people I know). In my experience, undergrad diff geom courses typically have students with a wide range of backgrounds & abilities, and getting comfortable with forms is a large hurdle for many. You can do everything that needs to be done with classical vector analysis. Forms are essential for abstract manifolds, which is what my books are about. $\endgroup$ – Jack Lee Jul 24 '14 at 23:48

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