Reddit mentions of Introduction to Smooth Manifolds (Graduate Texts in Mathematics, Vol. 218)

i have three categories of suggestions.

advanced calculus

these are essentially precursors to smooth manifold theory. you mention you have had calculus 3, but this is likely the modern multivariate calculus course.

• advanced calculus: a differential forms approach by harold edwards
  
  
• advanced calculus: a geometric view by james callahan
  
  
• vector calculus, linear algebra, and differential forms: a unified approach by john hubbard
  
  out of these, if you were to choose one, i think the callahan book is probably your best bet to pull from. it is the most modern, in both approach and notation. it is a perfect setup for smooth manifolds (however, all of these books fit that bill). hubbard's book is very similar, but i don't particularly like its notation. however, it has some unique features and does attempt to unify the concepts, which is a nice approach. edwards book is just fantastic, albeit a bit nonstandard. at a minimum, i recommend reading the first three chapters and then the latter chapters and appendices, in particular chapter 8 on applications. the first three chapters cover the core material, where chapters 4-6 then go on to solidify the concepts presented in the first three chapters a bit more rigorously.
  
  smooth manifolds
  
  
• an introduction to manifolds by loring tu
  
  
• introduction to smooth manifolds by john m. lee
  
  
• manifolds and differential geometry by jeffrey m. lee
  
  
• first steps in differential geometry: riemannian, contact, sympletic by andrew mcinerney
  
  out of these books, i only have explicit experience with the first two. i learned the material in graduate school from john m. lee's book, which i later solidifed by reading tu's book. tu's book actually covers the same core material as lee's book, but what makes it more approachable is that it doesn't emphasize, and thus doesn't require a lot of background in, the topological aspects of manifolds. it also does a better job of showing examples and techniques, and is better written in general than john m. lee's book. although, john m. lee's book is rather good.
  
  so out of these, i would no doubt choose tu's book. i mention the latter two only to mention them because i know about them. i don't have any experience with them.
  
  conceptual books
  
  these books should be helpful as side notes to this material.
  
  
• div, grad, curl are dead by william burke [pdf]
  
  
• geometrical vectors by gabriel weinreich
  
  
• about vectors by banesh hoffmann
  
  i highly recommend all of these because they're all rather short and easy reads. the first two get at the visual concepts and intuition behind vectors, covectors, etc. they are actually the only two out of all of these books (if i remember right) that even talk about and mention twisted forms.
  
  there are also a ton of books for physicists, applied differential geometry by william burke, gauge fields, knots and gravity by john baez and javier muniain (despite its title, it's very approachable), variational principles of mechanics by cornelius lanczos, etc. that would all help with understanding the intuition and applications of this material.
  
  conclusion
  
  if you're really wanting to get right to the smooth manifolds material, i would start with tu's book and then supplement as needed from the callahan and hubbard books to pick up things like the implicit and inverse function theorems. i highly recommend reading edwards' book regardless. if you're long-gaming it, then i'd probably start with callahan's book, then move to tu's book, all the while reading edwards' book. :)
  
  i have been out of graduate school for a few years now, leaving before finishing my ph.d. i am actually going back through callahan's book (didn't know about it at the time and/or it wasn't released) for fun and its solid expositions and approach. edwards' book remains one of my favorite books (not just math) to just pick up and read.

Lee is still the easiest and best for self study. https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395/ref=sr_1_3?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-3

followed by:

https://www.amazon.com/Introduction-Smooth-Manifolds-Graduate-Mathematics/dp/1441999817/ref=sr_1_2?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-2

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It's long, taking almost two volumes to get to Stokes theorem. If conciseness is important, you can just read Warner:

https://www.amazon.com/Foundations-Differentiable-Manifolds-Graduate-Mathematics/dp/0387908943/ref=sr_1_1?keywords=Warner+manifolds&qid=1558265966&s=gateway&sr=8-1

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Tu looks good, but I haven't read it carefully.

Anti-disclaimer: I do have personal experience with all the below books.

I really enjoyed Lee for Riemannian geometry, which is highly related to the Lorentzian geometry of GR. I've also heard good things about Do Carmo.

It might be advantageous to look at differential topology before differential geometry (though for your goal, it is probably not necessary). I really really liked Guillemin and Pollack. Another book by Lee is also very good.

If you really want to dig into the fundamentals, it might be worthwhile to look at a topology textbook too. Munkres is the standard. I also enjoyed Gamelin and Greene, a Dover book (cheap!). I though that the introduction to the topology of R^n in the beginning of Bartle was good to have gone through first.

I'm concerned that I don't see linear algebra in your course list. There's a saying "Linear algebra is what separates Mathematicians from everyone else" or something like that. Differential geometry is, in large part, about tensor fields on manifolds, and these are studied by looking at them as elements of a vector space, so I'd say that linear algebra is something you should get comfortable with before proceeding. (It's also great to study it before taking quantum.) I can't really recommend a great book from personal experience here; I learned from poor ones :( .

Also, there are physics GR books that contain semi-rigorous introductions to differential geometry, even if these sections are skipped over in the actual class. Carroll is such a book. If you read the introductory chapter and appendices, you'll know a lot. On the differential topology side of things, there's Schutz, which is a great book for breadth but is pretty material dense. Schwarz and Schwarz is a really good higher level intro to special relativity that introduces the mathematical machinery of GR, but sticks to flat spaces.

Finally, once you have reached the mountain top, there's Hawking and Ellis, the ultimate pinnacle of gravity textbooks. This one doesn't really fall under the anti-disclaimer from above; it sits on my shelf to impress people.

In all seriousness, the applications of analysis to geometry can be really interesting and insightful, but to get to them you would have to first have background in differential topology, which it seems you lack. That might be a good subject to start with. A good book would be John Lee's An Introduction to Smooth Manifolds.

Hey I'm a physics BSc turned mathematician.

I would suggest starting with topology and functional analysis. Functional analysis is the foundation of quantum mechanics, and topology is necessary to properly understand manifolds, which are the foundation of relativity.

I would suggest Kreyszig for functional analysis. It's probably the most gentle functional analysis book out there.

For topology, I would suggest John Lee. This topology text is unique because it teaches general topology with a view towards manifolds. This makes it ideal for a physicist. If you want to know about Lie algebras and Lie groups, the sequel to this text discusses them.

Study what you find the most interesting!

Does your linear algebra include the spectral theorem or Jordan canonical form? IMHO, a pure math subject that is relatively the easiest to learn and is useful no matter what you do is linear algebra.

Group theory (representation theory) has also served me well so far.

If you want to learn GR and Hamiltonian mechanics in-depth, learning smooth manifolds would be a must. Smooth manifolds are basically spaces that locally look like Euclidean spaces and we can do calculus on. GR is on a pseudo-Riemannian manifold with changing metric (because of massive stuffs). Hamiltonian mechanics is on a cotangent bundle, which is a symplectic manifold (whereas Lagrangian mechanics is on a tangent bundle.) John Lee's book is a gentle starting point.

Edit: If you feel like the review of topology in the appendix is not enough, Lee also wrote a book on topological manifolds.

Both Lee's and Tu's books are on my reading list. They both seem excellent.

However, my vote is for Professor Tu's book, mainly because it manages to get to some of the big results more quickly, and he evidently does so without a loss of clarity. In the preface to the first edition, he writes "I discuss only the irreducible minimum of manifold theory that I think every mathematician should know. I hope that the modesty of the scope allows the central ideas to emerge more clearly." Consequently, his book is roughly half the length of Lee's.

I'd rather hit the most essential points first, and then if I want a more expansive view, I'd pick up Lee.

Disclaimer: I may not participate very frequently, as I have some other irons in the fire, so you might want to weigh my vote accordingly. If your sub sticks around for a while, I'd definitely like to join in when I can.

I suggest either Tu or (easy) Lee.