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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.

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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. –  Ryan Budney Feb 9 '13 at 0:29
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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. –  Deane Yang Feb 9 '13 at 6:13
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6 Answers

up vote 9 down vote accepted

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.

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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. –  anon Feb 9 '13 at 1:31
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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. –  Sönke Hansen Feb 9 '13 at 11:44
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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.

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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. –  Ryan Budney Mar 12 '13 at 23:42
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Yes. This condition is satisfied by do Carmo's book(s). –  Alexandre Eremenko Mar 13 '13 at 1:40
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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.

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Sternberg's Lectures on Differential Geometry (AMS Chelsea) are wonderful and treat more than "just" Riemannian geometry.

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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...

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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.

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