Calculus on Submanifolds Challenge

Two multi-part 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 non-empty subset. The following are equivalent. i) M is a C^k E-submanifold 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 E-local 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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