## 22M:133:AAA Introduction to Smooth Manifolds 10:30A - 11:20A MWF 210 MLH (Spring 08)

Syllabus
PRACTICE EXAM

Week 1 (Jan 22 - Jan 25): Boothby Chapter I

Week 2 (Jan 28 - Feb 1): Boothby Chapter I Manifolds

HW 1 (due Feb 1) I.3: 1, 3; I.4:1, 4; I.5: 3, 4

Week 3 (Feb 4 - Feb 8): Boothby II.1

2.1

2.1 part 2

2.2

HW2 (due Feb 8) II.1: 2, 8

Week 4 (Feb 11 - 15): Boothby

2.3

2.4

2.5, 2.6

HW 3 (due Friday Feb 15)
HW3 part 1

HW3 part 2

Boothby II.5.1

Week 5 (Feb 18 - 22):

Boothby 2.7

Boothby 3.1 = Randell 1.1

HW 4 (due Friday Feb 22)

HW4 part 1

Boothby III.1.1

Week 6 (Feb 25 - March 1):
Boothby 3.2: RP^n, see also Randell example 1.1.11

Boothby 3.3 = Randell 1.2: Differentiable Functions

Randell 1.3: Lie groups

HW 5 (due Friday March 1)

Handout Problem 1: f is smooth implies f is continuous.

Boothby Problem III.1.4, Randell, Problem 1.4.2, Randell, Problem 1.4.3.

Week 7 (March 3 - 7): Tangent Space, Randell 2.1 (see also Boothby II.4, note similarity between proof of Thm 2.1.6 in Randell and proof of Thm 4.1 in Boothby; For another perspective, see Boothby 4.1)
HW 6 (due Friday March 7)

Week 8 (March 10 - 15): The differential map, Randell 2.1

Exam 1
Tuesday March 11

Definitions

Answers to Randell's exam 1

Spring Break (March 17 - 21)

Week 9 (March 24 - 28): review, The differential map, Randell 2.1, Boothby III.4

HW 7 (due Friday March 28)

1.) Calculate the standard basis for R^3.

2.) Calculate dg_p where g: RP^2 --> R, g([x, y, z]) = (x^2 + y^2)/||(x, y, z)||^2 using

(a) by using Randell's thm 2.1.10

(b) from the definition.

Week 10 (March 31 - April 4): Randell 2.2 - 2.4

HW 8 (due 4/4): Randell 2.5.1, 2.5.2, 2.5.3, 2.5.5 i, iv

Week 11 (April 7 - April 11):
Randell 2.3, 2.4.1,
defn,
linear algebra review,
S^n charts,
defns

HW 9 (due 4/14) Boothby III.4 #2, 3, 6; Randell 2.5.6

Week 13 (April 21 - April 25):
Randell Chapter 3,
flows,
vector fields

Hitchin Differentiable
Manifolds
Chapter 1

HW 10 (due 4/25) Randell 3-2, 3-3

Week 14 (April 28 - May 2): immersion review,

Exam 2

Week 15 (May 5 - May 9): tensors,

HW 13 (due 5/9) 2 problems from exam 2.

Final Exam week (May 12 - 16):

Final Exam 2:15 P.M. Monday, May 12, 2008