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I have just finished the book "Manfredo P. do Carmo - Differential Geometry of Curves and Surfaces". My aim is to reach to graduate level to do research, but articles are not only too advanced to study after Carmo's book, but also I don't think that they are readable by just studying Carmo's book at all for a self-learner like me. Please someone tell me a book for Differential Geometry more advanced than Carmo's book but readable esp. for self-learning . Thanks a lot.

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    $\begingroup$ What about do Carmo's "Riemannian Geometry" (which is, in some sense, a sequel)? The book covers some of the foundational material in Riemannian geometry that you would need to study modern Riemannian geometry and research papers in the field. After, that there are a number of possible directions you could take, which I would be happy to note if you are interested. $\endgroup$ – Amitesh Datta May 30 '15 at 0:37
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    $\begingroup$ Many people recommend Introduction to Smooth Manifolds by Lee. $\endgroup$ – littleO May 30 '15 at 0:40
  • $\begingroup$ Hi AlphaE, yes that's it (I wrote the full title in my previous comment). By the way, as @littleO sort of suggested, there are a number of directions other than differential geometry which you could take. Lee's book is (if I remember correctly) on the general theory of topological manifolds and probably covers some algebraic topology. So, if you are interested in algebraic topology, you could read that, or a number of other references too. (Actually, I think that knowledge of the fundamental group and differential forms/de Rham cohomology is probably necessary for "Riemannian Geometry" too.) $\endgroup$ – Amitesh Datta May 30 '15 at 0:57
  • $\begingroup$ @Amitesh Datta Hi, the reference for graduate level (that I need to cover it as a first direction to go) is "W M Boothby - An Introduction to Differentiable Manifolds and and Riemannian Geometry" which is not readable despite its appearance! Does do Carmo's "Riemannian Geometry" or Introduction to Smooth Manifolds by Lee cover that book? $\endgroup$ – L.G. May 30 '15 at 1:08
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    $\begingroup$ Hi @AlphaE, I read Boothby's book (that's where I first learnt about differentiable manifolds); I thought it was quite a well-written book. (I'd be curious to know why you think otherwise.) I think do Carmo summarizes a lot of the elementary material that he needs (much of which would be covered in more detail in Boothby's book, for example) in Chapter 0. I don't know how practical it would be to learn this material directly from Chapter 0 of do Carmo's book, though; it depends on your mathematical maturity. $\endgroup$ – Amitesh Datta May 30 '15 at 1:19
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Here's my answer to this question at length.

http://www.geometry.org/tex/conc/differential_geometry_books.html

In summary, if you are looking at the pure mathematics style of DG, you would be looking at do Carmo's "Riemmanian geometry", three books by John M. Lee, and for something more advanced, the books by Serge Lang, Bishop and Crittenden and Peter Petersen. And for the really advanced level, there's Schoen and Yau "Lectures on Differential Geometry", which lists many hundreds of open problems to work on at the postgraduate level.

There are also a few items on this web site which address the same question, some of them several years ago.

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  • $\begingroup$ I would also suggest O'Neill's book Semi-Riemannian Geometry with Applications to Relativity. He develops the theory in suitable generality to do general relativity and then devotes several chapters to FRW cosmology and black holes. $\endgroup$ – Neal May 30 '15 at 1:24
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    $\begingroup$ Nice list. I need to bookmark this. I would add one for the sake of physics. Many of my peers in theoretical physics studied Anomalies in Quantum Field Theory by Bertlemann. It has about 100 pages of pure math at the start and is one of the more lucid birds-eye views you'll find in the physics literature. $\endgroup$ – James S. Cook May 30 '15 at 2:09
  • $\begingroup$ James, that might be something for me to look up in the library. It's selling for 141.00 right now, used, or 199.00 new, on a certain very well known online bookstore. There are also other physics books which have excellent birds-eye view DG summaries, like Barrett O'Neill "The geometry of Kerr black holes", which I recently acquired for $20 (from the same well known online bookseller). However, I was guessing that the question was about the pure mathematical style of DG. My web page subdivides DG books into the mathematics style and the physics style. They have different objectives of course. $\endgroup$ – Alan U. Kennington May 30 '15 at 2:29

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