I realize that this may be a very general question, perhaps even an unclear one (if it is I apologize), but as someone looking for the best way to start learning about these topics, I find that there is no clear path to learning Differential Geometry / Differential Topology, as there is with Analysis or General Topology, or even Abstract Algebra
For example in Analysis, most agree that Principles of Mathematical Analysis by Walter Rudin is the place to begin, for Topology, Munkres book is the standard reference, and for Algebra, most tend to use either Dummit and Foote, Artin, Fraleigh or Lang.
For Differential Geometry/Differential Topology, I find that there are no standard texts, the only one I know of is Lee's Introduction to Smooth Manifolds, however I feel I currently lack the prerequisites to tackle that book properly.
Now I understand that to recommend a book to someone, you would need some gauge of their mathematical ability/maturity, but it is next to impossible to demonstrate that, so instead I can give a list of books that I'm currently reading through, and plan to read through in the next 3-6 months.
What I'm currently reading
- Principles by Mathematical Analysis (Baby Rudin)
- Linear Algebra Done Right (by Sheldon Axler)
- Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (by Hubbard and Hubbard)
What I plan on reading soon
- Calculus on Manifolds by Spivak
- Topology by Munkres
- Complex Analysis by Alfhors
- Abstract Algebra by Dummit and Foote
But after that I'm lost as to where to go further. I'm lost between Analysis on Manifolds by Munkres, A Comprehensive Introduction to Differential Geometry by Spivak, and do Carmo's Differential Geometry of Curves and Surfaces.
Or should I just skip all those intermediate books and go straight to Lee's Introduction to Smooth Manifolds?
A Side note I find that the more challenging a book I read is, and the more I struggle through a book, I develop a deeper understanding of the topics in the book, and a greater appreciation of the subject I'm studying as a whole. Based on the books I've read/plan to read, please recommend books that are not easy, but difficult and challenging.