# Book for Undergrad Differential Geometry

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?

• 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 – Ashley Jul 24 '14 at 22:29
• Spivak's A Comprehensive Introduction to Differential Geometry is never a bad choice. – rogerl Jul 24 '14 at 22:33
• 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. – Brad Jul 24 '14 at 22:50
• @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? – user166203 Jul 24 '14 at 22:51
• @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. ;) – user166203 Jul 24 '14 at 22:57

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.