I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is insufficient.
19 Answers
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!)

11$\begingroup$ I just want to point out that neither of these suggestions are actually about differential geometry, they cover only differential topology. Where differential topology is the study of smooth manifolds and smooth maps between them differential geometry is the study of linearalgebraic structures on smooth manifolds, which endow it with notions like length, area, angle, etc. $\endgroup$ Commented May 3, 2016 at 14:49

3$\begingroup$ Guillemin and Pollack, and Milnor work concretely on subsets of Rˆn and are not the best references if you need to learn modern machinery; you might find essential concepts there, and learn them fast, but not enough motivation for, e.g., forms without the demands of greater generality. $\endgroup$– rfabbriCommented Aug 18, 2016 at 19:35

1$\begingroup$ I believe that Guillemin and Pollack's book was based on Minor's but targeting undergraduates. $\endgroup$ Commented Apr 9, 2020 at 20:18
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, CheegerEbin. 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 CheegerGromoll 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 3manifolds.
For a more analysisoriented 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 coordinates 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.

6$\begingroup$ Definitely second Milnor's Morse Theory. After several failed attempts, that's what really made differential geometry make some sense to me. $\endgroup$ Commented Oct 18, 2009 at 2:01

6$\begingroup$ Definitely third Milnor's Morse Theory, at least the chapter on connections and Riemannian metrics (which is all I've read). And I wish more authors would take your advice about local coordinates and Christoffel symbols. $\endgroup$ Commented Oct 18, 2009 at 2:41

4$\begingroup$ I REALLY have to disagree about local coordinates. Its really hard to understand the coordinate free aspects without understanding how differential geometry works in Euclidean space first. Secondlyyou'll be incapable of speaking to physicists without understanding coordinates and general tensors and this is really one of the most important applications of manifolds there is. $\endgroup$ Commented Jul 13, 2021 at 5:46

1$\begingroup$ It took centuries to the most brilliant minds to find the proofs. If you can understand them by reading, you have done it well enough. $\endgroup$ Commented Feb 24, 2022 at 21:22

$\begingroup$ @FCardelle, not sure what you mean. I would say that it's impossible to learn any substantial mathematical subject, including different geometry, by just reading books about it. $\endgroup$ Commented Feb 24, 2022 at 21:34
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, hcobordism theorem, framed manifolds, and surgery on manifolds.
For the basic definitions, I very much like do Carmo's "Riemannian Geometry" (also on Google Books).
Once you have seen the basics, Bott and Tu's "Differential Forms in Algebraic Topology", which is one of the great textbooks, might be a nice choice. The best way to solidify your knowledge of differential geometry (or anything!) is to use it, and this book uses differential forms in a very handson way to give a clear account of classical algebraic topology. It wouldn't be a good first book in differential geometry, though.

7$\begingroup$ I want to second do Carmo's book  it's a really beautiful source, and unlike many introductory books on Riemannian geometry it actually proves some deep theorems (like the sphere theorem). $\endgroup$ Commented Oct 20, 2009 at 3:40
How basic is "basic"?
For the "basic" material I like the book "Introduction to Smooth Manifolds" by John Lee very much. It's very friendly and very accessible and nicely explains the ideas. Spivak's "Comprehensive Introduction to Differential Geometry" is also very nice, especially the newer version with nonugly typesetting. Warner's book "Foundations of Differentiable Manifolds and Lie Groups" is a bit more advanced and is quite dense compared to Lee and Spivak, but it is also worth looking at, after you become more comfortable with the basic material.

$\begingroup$ "How basic is "basic"?" Well that is a good question. I want to be able to converse and understand the essential material, but I'm not looking to become an expert. I would like to know enough to read about Lie groups and symplectic geometry without the differential geometry being an obstacle. $\endgroup$– GMRACommented Oct 14, 2009 at 19:55

2$\begingroup$ Lee's book is probably your best bet, then. Lee covers the rudiments quite nicely, and then also gets into some basic symplectic geometry and Lie groups. $\endgroup$ Commented Nov 10, 2009 at 23:52
I feel like this needs to be asked: Is there really such a subject as "basic differential geometry?" I don't believe there is.
Consider the following list of standard topics in "differential geometry" that are, depending on the prof's research interests, either absolutely essential or not covered at all in an intro grad course: partitions of unity, differential forms and Stokes Theorem, Frobenius Theorem, basic Riemannian geometry, fiber bundles, index of a vector field, Lie groups, etc. To wit, if you look at many of the texts listed by other posters, many of them have essentially NO overlap. Just about the only thing that is indisputably a necessary topic in a differential geometry course is the definition of a smooth manifold (and maybe the inverse/implicit function theorem, but probably not). Ironically, even the definition of a smooth manifold is not even a point of intersection, since one may choose to work purely with submanifolds of R^n.
In short, if someone wants to learn some differential geometry, one first has to decide "what kind" or for "what purpose."
Here is my list of about 60 books and historical works about differential geometry.
differential geometry textbooks
About 50 of these books are 20th or 21st century books which would be useful as introductions to differential geometry at this point in time. In this list, I give some brief indications of the contents and suitability of most of these books.
I would recommend Barrett O'Neill's SemiRiemannian Geometry. It's very good. The latter chapters concern general relativity, but the earlier chapters are purely mathematical and contain lots of nice differential geometry.

1$\begingroup$ +1 for an outstanding book that doesn't get enough attention and would be great reading for either math or physics graduate students. UCLA's Peter Petersena guy who knows a thing or three on the subjectfirst learned the subject from the book and its legendary author. $\endgroup$ Commented Aug 27, 2016 at 1:49

1$\begingroup$ +1 — the resident string theory professor at my college gave me his copy of this book when I asked about general relativity and it has served me incredibly well. (judging by the massive number of annotations it served him well too!) $\endgroup$ Commented Mar 13, 2018 at 21:32

$\begingroup$ A great textbook, that I discovered only recently $\endgroup$ Commented Apr 11, 2020 at 16:24
For a basic undergraduate introduction to differential geometry, I'd highly recommend Manfredo Do Carmo's Differential Geometry of Curves and Surfaces. Nicely done and very approachable, and you'd be well prepared to tackle Spivak's books next.
Noel J.Hicks lectures in differential geometry http://www.maths.ed.ac.uk/~aar/papers/hicks.pdf
For a very detailed presentation of some topics from the basic theory in the spirit of algebraic geometry, see Brian Conrad's handouts.
I would strongly recommend 'An introduction to manifolds' and 'Differential geometry' by Loring Tu. Compared to most other books mentioned, these are recently published. Having gone through both of them, I can vouch for the clarity of presentation and readability. Required prerequisites are minimal, and the proofs are well spelt out making these suitable for self study. The exercises are nice too.
"Lectures on Differential Geometry" by Chern, Chen, and Lam is an excellent book, and one which truly addresses differential geometry rather than differential topology alone.
I would like to suggest the three volume set by Dubrovin, Fomenko, Novikov (Modern GeometryMethods and Applications) as a supplementary reference. They have a somewhat unique style and approach to the subject. The first volume begins with surfaces, the second volume goes on to manifolds. They give examples from physics along the way, which some may find interesting/useful.
I also like the chatty, informal style of M. Berger. He doesn't shy away from giving informal descriptions of ideas and motivations behind definitions. Perhaps most books try to do this, but Berger is particularly generous with it, and good at it, in my opinion. I have his book A Panoramic View of Riemannian Geometry in mindthis may not be the best place to learn about differential geometry for the first time, but I think some of his insights/comments would be useful even for beginners (see, e.g., pp 143151). I think he has other, more elementary books on geometry but I don't have the references right now.
My take on it is like this:
Smooth manifolds
 Loring Tu, Introduction to manifolds  elementary introduction,
 Jeffrey Lee, Manifolds and Differential geometry, chapters 111 cover the basics (tangent bundle, immersions/submersions, Lie group basics, vector bundles, differential forms, Frobenius theorem) at a relatively slow pace and very deep level.
 Will Merry, Differential Geometry  beautifully written notes (with problems sheets!), where lectures 127 cover pretty much the same stuff as the above book of Jeffrey Lee
Basic notions of differential geometry
Jeffrey Lee, Manifolds and Differential geometry, chapters 12 and 13  center around the notions of metric and connection.
Will Merry, Differential Geometry  lectures 2853 also center around metrics and connections, but the notion of parallel transport is worked out much more thoroughly than in Jeffrey Lee's book.
Sundararaman Ramanan, Global calculus  a highbrow exposition of basic notions in differential geometry. A unifying topic is that of differential operators (done in a coordinatefree way!) and their symbols.
What I find most valuable about these books is that they try to avoid using indices and local coordinates for developing the theory as much as possible, and only use them for concrete computations with examples.
However, the above books only lay out the general notions and do not develop any deep theorems about the geometry of a manifold you may wish to study. At this point the tree of differential geometry branches out into various topics like Riemannian geometry, symplectic geometry, complex differential geometry, index theory, etc., not to mention a close sister, differential topology.
I will only mention one book here for the breadth of topics discussed
 Arthur Besse, Einstein manifolds  reviews Riemannian geometry and tells about (more or less) the state of the art at 1980 of the differential geometry of Kähler and Einstein manifolds.

1$\begingroup$ Thank you for the reference to Will Merry's notes, they look great. $\endgroup$– seubCommented Apr 18, 2020 at 22:05
Another book I find pretty readable is Modern Differential Geometry for Physicists by Chris Isham. Don't worry about the "physicists" bit in the title, the proofs are not missing there :)

1$\begingroup$ This book helped me to learn connections on principal bundles.. $\endgroup$ Commented Sep 23, 2019 at 4:23
I am currently using Elementary Differential Geometry by Barrett O'Neill. So far it has been a good reference so long as you have a good grasp what was taught in most linear algebra courses. I will have to update this as the course continues an compare it to the other texts mentioned above.
S. Sternberg's recent Curvature in Mathematics and Physics is at about the same level as some of the other suggestions, but includes some extra material hard to find in textbooks at this level. It's also available from Dover, so quite inexpensive.
I would like to add the following notes by Nigel Hitchin: https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Differentiable_manifolds/manifolds2014.pdf