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For quite a while I have been interested in differential geometry. In 1-2 months I might find myself with some time on my hands. I have heard and cursorily flipped through a few of them but I would really appreciate a more informed opinion. If anyone would comment about the style, prerequisites, etc of these books I would be grateful.

  1. Semi-Riemannian Geometry with applications to Relativity (Barret O'Neill)
  2. Elementary Differential Geometry (Barret O'Neill)
  3. The Geometry of Physics (Theodore Frankel)
  4. Differential Geometric Structures (Walter Poor)
  5. A Comprehensive Course of Differential Geometry (Spivak) (Vol 1 & 2 Maybe?)
  6. Tensor analysis on Manifolds (Bishop & Goldberg)
  7. Differential Forms & Connections (RWR Darling)

I like what I have seen of 1,2 & 3.

1 seems to be the coolest book of the lot. 2 seems to be the standard barring do carmo. 3 covers a LOT but it isn't all that rigorous.

Do I need to read 2 before I read 1? Because 1 seems to be pretty wild stuff like singularity theorems and since they're both written by the same author so is 2 a prerequisite for 1 or I can jump in directly?

For some background, I have read Hubbard's Vector calculus book and Linear algebra done right by axler and linear algebra by strang. I have also worked through the first few chapters of apostol's mathematical analysis book so I am not new to rigor.

Falgun Sukhija
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  • If you do not know anything about differential geometry I recommend that you start with a book like Elementary Differential Geometry from Christian Bär. It's fun to read and gives you an intuition about the basic concepts. After that I recommend Introduction to Smooth Manifolds from John Lee. Learning Differential Geometry and later Riemannian Geometry or Symplectic Geometry is like learning a language and if you start with advanced topics you might get lost. – Philipp Jul 08 '21 at 17:36
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    As you are already familiar with real analysis, I would recommend you Differential geometry of curves and surfaces from Manfredo do Carmo. I used it in my first course of differential geometry and I can say it is good book to start. The proofs are well explained and the author make sure to present several examples. Some challenging exercises too. – Gio Jul 08 '21 at 17:48
  • @Cornman I'd like to very much but right now I am trying to sink my teeth into baby rudin – Falgun Sukhija Jul 08 '21 at 18:12
  • @nmasanta I had read through this post and several others before posting this . I guess my main question is whether I can directly start reading 1 before I read 2. It seems by far the coolest book in the lot. Also personalized comments by someone who went through these texts on their own would help – Falgun Sukhija Jul 09 '21 at 02:45
  • Another book to be aware of is Tristan Needham's new book Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts. Needham's previous book Visual Complex Analysis is very popular. – littleO Jul 09 '21 at 03:06

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