Calculus on Submanifolds Challenge
Two multipart questions about submanifolds given to my class as challenges based on the book Calculus on Manifolds by Spivak. The more detailed the better.

What is the formal definition of parameterization you are using in class?

Hey, give me a minute to type it up. Is there anyway I could send a pdf to you? If not there should not be any issue with using wichever definition you want as long as you announce it first.

Well the issue is usually my definition would be a local C^\infty diffeomorphism \phi from an open U in R^n onto an open V inside of M, but this makes the first part of the problem trivial. Maybe you can just type it here? If you know latex I can just read it when it's uncompiled.

If you're using a textbook you could just refer me to the definition so that what I write will be consistent with the notions you are familiar with.

Oh, got it. Here goes: Let E, G be Banach spaces and M ⊂ G a nonempty subset. The following are equivalent. i) M is a C^k Esubmanifold of G. ii) For each p ∈ M, there exists a embeddening C^k ϕ : U ⊂ E → G (U is open) such that p ∈ ϕ(U) ⊂ M and ϕ(U) is open of M in the induced topology. ϕ is called a C^k Elocal parameterization of M in p.

Great, got it. Can I assume that all manifolds are C^\infty manifolds?

Just checked, we proved that after the challenge was issued. So no, can't assume, unfortunately.

Okay finally: have you proved the inverse function theorem on Banach spaces?

Yes, we did.

Hey, just checking on this. Doing alright? Will you be able to finish it in time?

Any other info you need, just ask.

Yes, it's pretty easy. However I just lost all my progress when the website reloaded so I'll type it up somewhere else and upload it in a couple hours.

As I submitted I noticed a typo: in 1.d.) I write "the embedded close submanifold U on which x_2^2 + y_2^2" but this should read "... on which x_2^2 + y_2^2 = 1". Somehow that got left out.

Another important thing to note: in 2.a.) the conditions "cos(a) \neq 0" and "sin(a) \neq 0" should be switched, that was a typo.

Finally: "parallelization" in 2.c.) is more accurately a local basis of sections at every point, it is not a parallelization.
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