## GEOMETRY 2, FALL 2012

See also the Study Information System
SIS page and (requires login)
Absalon

**MANDATORY EXERCISE ASSIGNMENT:**
here (the return deadline, January 8, has passed).
Solutions
here
### LECTURE NOTES

Differentiable Manifolds, by Henrik Schlichtkrull. 2012 version.

ISBN 978-87-7078-283-8, Polyteknisk Boghandel.
### LECTURES

19/11 Monday 1.1-1.3

20/11 Tuesday 1.4-1.6

23/11 Friday 1.7-1.8

26/11 Monday 2.1-2.4

27/11 Tuesday 2.4-2.7

30/11 Friday 2.8-2.10

3/12 Monday 3.1-3.5

4/12 Tuesday 3.6-3.8

7/12 Friday 3.9-3.10

10/12 Monday 4.1-4.3

11/12 Tuesday 4.4-4.6

14/12 Friday 4.7-4.10

17/12 Monday 5.1-5.5

18/12 Tuesday 5.6-5.8

21/12 Friday 5.9-5.11

4/1 Friday The lecture is replaced by a mandatory written assignment (see below)

7/1 Monday 6.1-6.4

8/1 Tuesday 6.5-6.8

11/1 Friday 6.9-6.10

22/1 Tuesday. Question session (Aud 8, starts 13:15).

### EXERCISE CLASSES

Exercise classes are used for problem solving,
and for presentation of exam questions by participants
as preparation for the oral exam.
There are 12 problem programs P1-P12, and 12 exam questions E1-E12.
The plan below may be revised when the number of students is known -
the goal is that every student presents one question.

20/11 Tuesday Exercises cancelled (but not lectures). They are replaced
by extra classes up to the exam.

23/11 Friday P1

27/11 Tuesday P2

30/11 Friday P3

4/12 Tuesday P4+E1 (Anna)

7/12 Friday P5+E2 (Jens Jacob)

11/12 Tuesday P6+E3 (Jason)

14/12 Friday P7+E4 (Rie)

18/12 Tuesday P8

21/12 Friday P9+E5 (Ji Yoon)

4/1 Friday P10+E6 (David)

8/1 Tuesday P11

11/1 Friday P12+E7+E8 (Birger + Helle)

15/1 Tuesday E9+E10+E1 (Evgeny + Nikolaos + Konstantinos)

18/1 Friday E11+E12 (Thomas + Estibaliz)

### MANDATORY EXERCISE ASSIGNMENT

There is one mandatory written assignment during the course.
It will be web-posed Wednesday January 2, and must be returned by
Tuesday January 8 at the start of the exercise class.

### PROBLEM PROGRAMS

Select problems from the following list. The problems
cover the mentioned sections. Emphasis will be on the
highlighted problems.

P1: 1.1-1.4, Exercises 1.1-1.7.
3,4,5,7

P2: 1.5-1.8, Exercises 1.8-1.17.
11,12,13,15,17

P3: 2.1-2.5, Exercises 2.1-2.10.
4,5,6,7,9,10

P4: 2.6-2.10, Exercises 2.11-2.15.
All

P5: 3.1-3.6, Exercises 3.1-3.5.
2,3,4

P6: 3.7-3.10, Exercises 3.6-3.10.
6,7,8,10

P7: 4.1-4.4, Exercises 4.1-4.8.
2,3,4,5

P8: 4.5-4.8, Exercises 4.9-4.15.
10,11,12,13

P9: 5.1-5.6, Exercises 5.1-5.5.
2,3,4

P10: 5.7-5.10, Exercises 5.6-5.12.
8,9,10,11

P11: 6.1-6.5, Exercises 6.1-6.7.
1,3,4,6

P12: 6.6-6.9, Exercises 6.8-6.14.
10,12,13,14

### EXAM

23-24/1 Oral exam. Please see the schedule in ABSALON.

### EXAM QUESTIONS

E1: Manifolds in R^n

E2: Abstract manifolds. Projective space.

E3: The tangent space of an abstract manifold

E4: Smooth maps and their differentials

E5: Submanifolds

E6: The orthogonal group is a Lie group

E7: Partition of unity

E8: Embedding in Euclidean space

E9: Connectedness and components

E10: Vector fields and Lie brackets

E11: The Lie algebra of a Lie group

E12: The Lie algebra of GL(n,R)