Riemannian Geometry book to complement General Relativity course?

What would be a good Riemannian Geometry (or Differential Geometry) book that would go well with a General Relativity class (offered by a physics department)? I'm in one right now, but I'd like a pure math perspective on the math that's introduced as I can imagine, inevitably some things would be swept under the rug and I'd like a fuller picture. I'm looking at John M. Lee's "Riemannian Manifolds" and Jeffrey Lee's "Manifolds and Differential Geometry".

Are these books suitable? What parts should I study?

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Do you know Nakahara's book? –  Sigur Feb 5 '13 at 21:16
Nakahara is too terse for me. A lot of material thrown in the smallest amount of pages. But it is a good guide of the material I think. –  Sickell Feb 5 '13 at 21:28

2 Answers

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Thanks. The book by O'Neill looks great. –  Sickell Feb 5 '13 at 22:41
+1 for O'Neill. –  Neal Feb 5 '13 at 23:40

The new book by Sternberg (a freely available version is linked in the answer by Will Jagy) is very affordable and focused on just what you may need:

If you want something more detailed and explanatory than Nakahara's, as a good bridge to the more purely mathematical references, you should take a look at these excellent titles:

In particular I would recommend the new book by Eschrig to complement any general relativity or gauge theory courses, with the formal background; it is filled with geometric and visual motivation alongside the formal concepts and arguments. Nevertheless, these books do not focus on (pseudo)-Riemannian geometry per se, but on general differential geometry, trying to introduce as many concepts as possible for the needs of modern theoretical physics.

Most purely mathematical books on Riemannian geometry do not treat the pseudo-Riemannian case (although many results are exactly the same). That is why the books on "geometry for physicists", from easy to very formal level, are the best first approach to the mathematical background. Once you get into advanced general relativity, the most mathematical treatments are given by books like Wald, Hawking/Ellis, de Felice, Fré et al... where the physical study of the mathematics of pseudo-Riemannian geometry is explained. Check out the references mentioned in this other answer. You may also find useful this other answer on self-learning differential topology and differential geometry.

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Thanks for the references. I've had some exposure to manifolds and differential forms etc by reading parts of Nakahara and Lee and Munkres' books on the subject. The Eschrig book looks nice. On an unrelated note, I've been reading some of your posts and I have many general questions to ask you as my academic goals seem quite similar to yours except you have more experience. How can I contact you? –  Sickell Feb 6 '13 at 6:08
@Sickell, look for my profile at mathoverflow.net, you will find my email in the about me text box (It will appear until you contact me). Do not hesitate to ask anything! –  Javier Álvarez Feb 6 '13 at 9:28