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?

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.
