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