Math 250A

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

Homework Problems

pp.5-7 4,5,7,12,16,17

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.

p.55-56 1,3,9,10,11

Hand written lectures:

Lecture 2

Lecture 3

Lecture 4

Lecture 5

Lectures 6&7

Lecture 8

Lecture 9

Lecture 10

Lecture 11

Lecture 12

Lecture 13

Lecture 14

Lecture 15

Lecture 16

Lecture 17

Lecture 18

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

Lecture 19

Lecture 20

Lecture 21

Lecture 22

Lecture 23

Lecture 24

Lecture 25

Lecture 26

Lecture 27

Lecture 28

Lecture 29

Lecture 30