Reading list for basic differential geometry?

UPDATE:

I am teaching a graduate differential geometry course focusing on Riemannian geometry and have been looking more carefully at several textbooks, including those by Lee, Tu, Petersen, Gallot et al, Cheeger-Ebin. I also wanted to focus on differential geometry and not differential topology. In particular, I wanted to do global Riemannian geometric theorems, up to at least the Cheeger-Gromoll splitting theorem. So far, I like Petersen's book best.

Also, as it happens, Cheeger is teaching a topics course on Ricci curvature. He is relying on notes he has written, which I can recommend, at least for a nice overview of the subject. They lay the groundwork for his recent work on Ricci curvature. One of them, Degeneration of Riemannian metrics under Ricci curvature bounds, is available on Amazon.

OLD POST:

First, follow the advice that a former Harvard math professor used to give his students. He would point to a book or paper and say, "You should know everything in here but don't read it!". My interpretation of this is to look first at only the statements of the definitions and theorems and try to work out the proofs yourself. Peek at the book only as needed.

Second, follow the advice of another former Harvard professor and develop your own notation. Why? Because it appears that each differential geometer and therefore each differential geometry book uses its own notation different from everybody else's. So you'll go nuts, unless you have your own notation and you translate whatever you're reading into your own notation. Of course, this is a natural thing to do, while you're trying to work out your own proof anyway.

Spivack is for me way too verbose and makes easy things look too complicated and difficult.

I love Guillemin and Pollack, but it is just a rewrite for undergraduates of Milnor's "Topology from a Differentiable Viewpoint". And it's really about differential topology (that is the title after all) and not differential geometry.

For a really fast exposition of Riemannian geometry, there's a chapter in Milnor's "Morse Theory" that is a classic. The rest of the book is great, of course.

Another classic that ties in well with Lie groups is Cheeger and Ebin's "Comparison Theorems in Riemannian Geometry".

I'm recommending only older books, because I haven't kept up with all the newer books out there. One that I also really like is "Riemannian Geometry" by Gallot, Hulin, Lafontaine.

And, back in the day, many of us also learned a lot by reading Thurston's notes on 3-manifolds.

For a more analysis-oriented book, check out Aubin's "Some Nonlinear Problems in Riemannian Geometry". He has a book on Riemannian geometry, but I don't know it very well.

One piece of advice: Avoid using local co-ordinates and especially those damn Christoffel symbols. They have no geometric meaning and just get in the way. It is possible to do almost everything without them. The books I've recommended, except possibly Aubin, aim for this.

To Kevin's excellent list I would add Guillemin and Pollack's very readable, very friendly introduction that still gets to the essential matters. Read "Malcolm's" review of it in Amazon, I agree with it completely.

Milnor's "Topology from the Differentiable Viewpoint" takes off in a slightly different direction BUT it's short, it's fantastic and it's Milnor (it was also the first book I ever purchased on Amazon!)

I'd start with Lee's Introduction to Smooth Manifolds. It covers the basics in a modern, clear and rigorous manner. Topics covered include the basics of smooth manifolds, smooth vector bundles, submersions, immersions, embeddings, Whitney's embedding theorem, differential forms, de Rham cohomology, Lie derivatives, integration on manifolds, Lie groups, and Lie algebras.

After finishing with Lee, I'd move on to Hirsch's Differential Topology. This is more advanced then Lee and leans more towards topology. Also, the proofs are much more brief then those of Lee and Hirsch contains many more typos than Lee. The topics covered include the basics of smooth manifolds, function spaces (odd but welcome for books of this class), transversality, vector bundles, tubular neighborhoods, collars, map degree, intersection numbers, Morse theory, cobordisms, isotopies, and classification of two dimensional surfaces.

These two should get you through the basics. However, if that is not enough, I'd move on to Kosinski's Differential Manifolds which covers the basics of smooth manifolds, submersions, immersions, embeddings, normal bundles, tubular neighborhoods, transversality, foliations, handle presentation theorem, h-cobordism theorem, framed manifolds, and surgery on manifolds.