11
$\begingroup$

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.

Improve this question
$\endgroup$
3
  • 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 Budney
    Commented Feb 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 Yang
    Commented Feb 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) $\endgroup$
    – Chee
    Commented Mar 8, 2022 at 18:30

8 Answers 8

Reset to default
19
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • 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. $\endgroup$
    – anon
    Commented Feb 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. $\endgroup$
    – user80744
    Commented Feb 9, 2013 at 11:44
6
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • 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. $\endgroup$
    – Ryan Budney
    Commented Mar 12, 2013 at 23:42
  • 1
    $\begingroup$ Yes. This condition is satisfied by do Carmo's book(s). $\endgroup$
    – Alexandre Eremenko
    Commented Mar 13, 2013 at 1:40
3
$\begingroup$

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.

Improve this answer
$\endgroup$
3
$\begingroup$

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

Improve this answer
$\endgroup$
3
$\begingroup$

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...

Improve this answer
$\endgroup$
1
2
$\begingroup$

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.

Improve this answer
$\endgroup$
2
$\begingroup$

I'm studying differential geometry in order to learn general relativity. My advisor recommended the following book: Semi-Riemannian Geometry, With Applications to Relativity, by Barrett O'Neill.

https://www.amazon.com/Semi-Riemannian-Geometry-Applications-Relativity-Mathematics/dp/0125267401

Improve this answer
$\endgroup$
1
$\begingroup$

Start with Topology from the differentiable viewpoint by John Milnor.

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

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .