## GEOMETRY 2, FALL 2013

See also the
course catalogue and (requires login)
Absalon

### LECTURE NOTES

Differentiable Manifolds, by Henrik Schlichtkrull. 2013 version.

ISBN 978-87-7078-353-8, Polyteknisk Boghandel.
### LECTURES

(preliminary plan)

18/11 Monday 1.1-1.5

19/11 Tuesday 1.6-1.8

22/11 Friday 2.1-2.3

25/11 Monday 2.4-2.7

26/11 Tuesday 2.7-2.10

29/11 Friday The Grassmann manifold. Extra notes here

2/12 Monday 3.1-3.5

3/12 Tuesday 3.5-3.8

6/12 Friday 3.8-3.10

9/12 Monday 4.1-4.3

10/12 Tuesday 4.4-4.6

13/12 Friday 4.7-4.10

16/12 Monday 5.1-5.5

17/12 Tuesday 5.6-5.8

20/12 Friday 5.9-5.11

3/1 Friday No lecture. (See about a mandatory written assignment below)

6/1 Monday 6.1-6.4

7/1 Tuesday 6.5-6.8

10/1 Friday 6.9-6.10

14/1 Tuesday. Question session. AUD 6

### 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 (see below), and 13 exam questions E1-E13 (also below).
Further presentations will be planned once the exact number of
students is known - the goal is that every student presents one question.

The exercise classes are conducted by Wolfgang Steimle and by Massimiliano Ungheretti.

19/11 Tuesday P1

22/11 Friday P2

26/11 Tuesday P3

29/11 Friday P4

3/12 Tuesday P5+E1

6/12 Friday P6+E2

10/12 Tuesday P7+E3

13/12 Friday Selected exercises with which you were behind+E4. Note that the remaing P-program has been postponed by half a week.

17/12 Tuesday P8+E5

20/12 Friday P9+E6

3/1 Friday No class. (See about a mandatory written assignment below)

7/1 Tuesday P10+E7+E8

10/1 Friday P11+E9+E10

14/1 Tuesday P12+E11+E12+E13

### MANDATORY EXERCISE ASSIGNMENT

A mandatory written assignment replaces the teaching of January 3.
It will be web-posed January 2, and must be returned by
Tuesday January 7 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

The oral exam is scheduled for Friday 24, but one day is not enough.
Please see here

The scope of the exam is as follows: The student draws a question from the following
list, and is allowed approximately 30 minutes of preparation.
The examination lasts 25 minutes, of which 15-20 minutes will be spent on the presentation.
After that questions will be asked in the entire
curiculum of the course. At the end the student receives a grade.

The official language of the exam is English (but Danish will be accepted as well).

### EXAM QUESTIONS

E1: Manifolds in R^n

E2: Abstract manifolds. Projective space.

E3: The Grassmann manifold.

E4: The tangent space of an abstract manifold

E5: Smooth maps and their differentials

E6: Submanifolds

E7: The orthogonal group is a Lie group

E8: Partition of unity

E9: Embedding in Euclidean space

E10: Connectedness and components

E11: Vector fields and Lie brackets

E12: The Lie algebra of a Lie group

E13: The Lie algebra of GL(n,R)