I want to start studying differential geometry but I can't seem to find a proper starting path. Whenever I try to search for differential geometry books/articles I get a huge list. I know that it is a broad topic, but I want some advice for you regarding the books and articles. I want to learn differential geometry and especially manifolds. I have some background in abstract algebra, linear algebra, topology, real/complex analysis.

2$\begingroup$ I really like Takashi Sakai's "Riemannian Geometry" as well as the same titled book by Gallot. Marcel Berger's "Panoramic View" is great once you know the basics. $\endgroup$– Ryan BudneyFeb 9, 2013 at 0:29

3$\begingroup$ I second the recommendation of the book by Gallot, Hulin, and Lafontaine. I learned Riemannian geometry from the book by Cheeger and Even, Comparison Theorems in Riemannian Geometry, which is beautiful. You probably need to consult other books when reading it but it is a wonderful guide through the subject. $\endgroup$– Deane YangFeb 9, 2013 at 6:13

$\begingroup$ I like a lot Takashi Sakai's "Riemannian Geometry". This book has a strong analysis flavor and seems to be more difficult than Lee's book on "Introduction to Riemannian manifolds". If you have a lot of time, read M. Spivak's "A comprehensive introduction to differential geometry", which seems to be easier than the previous two. (Personal opinion though) $\endgroup$– CheeMar 8, 2022 at 18:30
8 Answers
I would recommend Lee's book "Introduction to Smooth Manifolds." It's a long book but is comprehensive, has complete proofs, and has lots of exercises.

4$\begingroup$ It's a good book, but it takes 327 pages to get to metrics. For someone wanting to learn differential <i>geometry</i>, there are faster routes. $\endgroup$– anonFeb 9, 2013 at 1:31

6$\begingroup$ I would rather advise to read J. Lee's book "Riemannian Manifolds  An introduction to curvature". It heads straight towards geometry, is very clear and readable, and is of moderate length. Some knowledge about manifolds is required to read the book, but not really much. $\endgroup$– user80744Feb 9, 2013 at 11:44
M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Wilmington, DL, 1979 is a very nice, readable book.
If you prefer something shorter, there are two books of M. Do Carmo, 1. Differential geometry of curves and surfaces, and 2. Riemannian geometry.

1$\begingroup$ Although I like the Spivak books for their massive content, I think someone who is learning differential geometry for the first time wants something that cuts more efficiently to the meat, making less of an attempt to be encyclopedic. $\endgroup$ Mar 12, 2013 at 23:42

1$\begingroup$ Yes. This condition is satisfied by do Carmo's book(s). $\endgroup$ Mar 13, 2013 at 1:40
I recommend an older book, Notes on Differential Geometry by Noel Hicks. What I like about it is that it starts with manifolds embedded in $R^n$, and shows how all of the concepts of differential geometry naturally arise there.
Sternberg's Lectures on Differential Geometry (AMS Chelsea) are wonderful and treat more than "just" Riemannian geometry.
Let's suppose you can either read Russian or French, I would recommend M.Postnikov's Lectures on Geometry 3 and 4, this is really the most coherent book I've read. Okay, it's a series,though...

2$\begingroup$ You can find the Postnikov lectures in English at urss.ru/cgibin/db.pl?page=Search $\endgroup$– 5spaceAug 15, 2014 at 6:46
I would go with John Lee's Introduction to smooth manifolds and back it up with doCarmo's Riemannian geometry. If you wish to delve further into Riemannian stuff, go for the classic Comparison theorems in Riemannian geometry by Cheeger and Ebin.
I'm studying differential geometry in order to learn general relativity. My advisor recommended the following book: SemiRiemannian Geometry, With Applications to Relativity, by Barrett O'Neill.
https://www.amazon.com/SemiRiemannianGeometryApplicationsRelativityMathematics/dp/0125267401
Start with Topology from the differentiable viewpoint by John Milnor.

1$\begingroup$ Or the slightly thicker book that assumes a little less of the reader "Differential Topology" by Guillemin and Pollack. $\endgroup$ Aug 21, 2021 at 5:49