I am interested in learning about manifolds using Lee's Introduction to Smooth Manifolds but I have some questions regarding its prerequisites.
I have done multivariable calculus (Colley's Vector Calculus), single-variable analysis (Rudin chapters 1-7), ODEs (Boyce), point set topology (Munkres) and linear algebra (Friedberg).
However, I have noticed that in the four appendices of the book, there is the implicit function theorem, which is a subject that is done in chapters 9 and 10 of Rudin's PMA. Also, since I haven't taken undergraduate differential geometry such as O'Neill or Do Carmo, I also lack knowledge regarding the subject.
Would it be possible to jump right into Introduction to Smooth Manifolds or would it be better to learn those two subjects beforehand?
Also, is Introduction to Topological Manifolds by the same author a prerequisite for the book or is it possible to do those two books concurrently?
Thank you.
Best Answer
I think you have almost all the prerequisites needed. I took a look at the contents of Colley's vector calculus; the only other thing you really need is the inverse and implicit function theorem (which are actually equivalent theorems), which is already proved in one of the appendices (but in Lee's Appendix, only the proof is given with very little further elaboration, because it's meant to be review). So, I suggest you learn the theorems, and do some practice questions from a multivariable calculus text (e.g. I loved Loomis and Sternberg for all of multivariable calculus. The IFTs are Chapter 3, section 11, and all those exercises are "standard"). Here's my personal warning: do not use Rudin (C9) for multivariable calculus review; you will struggle A LOT and learn very little.
The inverse and implicit function theorems are "non-linear extensions" of basic linear algebra results. After all, differential calculus is a theory of locally approximating by linear functions. So, if you keep this idea in mind, these theorems become much more tractable. There are several good answers on MSE which touch on the intuition of these theorems (so spend some time searching for them:)
Other than that, I think with your topology preparation (Munkres), and linear algebra (Friedberg), you have all the necessary requirements. I haven't read Introduction to TOpological Manifolds, and I was able to get by a good chunk of Introduction to Smooth Manifolds. By the way, on youtube you can find Lectures by Frederic Schuller which introduce the basics. Watching the first few lecutures (e.g lectures 2-6) will serve as a nice complement to reading the texts.