So far two misprints on the exam: In Problem 4 a) let X \in Vect(X) should say let X \in Vect(M) At the end of 4 c) it should say: \phi(x)=x for x not in U then \phi is a diffeomorphism of M)
Take home final exam
Below is a proof of the inverse function theorem that I took from the Harvard web page. It in turn is basically the proof in Spivak's Calculus on Manifolds.
The inverse function theorem
Below is a nice discussion of the long line by Richard Koch (U. Oregon).
The long line
The next file is chapter 1 in Warner's Differential Geometry:
Differential Geometry Chapter 1
p. 12 4,5
p. 18 3,5
p. 25 1,2,11,13 Read section 4 of chapter 1 in text.
p.50 Warner problems 6,9,10,14 (In Warner's book the second axiom is a part of the definition of a manifold. Additional Exercises:
1. Prove that if M and N are connected, compact manifolds and f: M -> N is smooth and has a bijective differential at every point then f defines a covering of N. (In particular f is surjective.)
2. Prove that if M is a connected one dimensional manifold with an everywhere non-vanishing vector field that M satsifies the second axiom of countability.
Below is a problem set that leads to a completion of our proof that a 1-dimensioal cnnected compact manifold is diffeomorphic with S^1.
More problems 11/20
Problems from the text.
The following is to expand on the argument in Lecture 18 that came out a bit scrambled.
An additional explanation for material in Lecture 18