Book for Undergrad Differential Geometry

Question

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?

Answer 1

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.

Answer 2

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.