# Øvingar (Exercises)

Exercise 1 (Week 3) Exercise set 1

Exercise 2 (Week 4) Exercise set 2

Exercise 3 (week 5) Exam 2010, Problem 1 and 4. Dundas Ex. 3.3.7.

Exercise 4 (week 6) Dundas : Exercise 3.4.15, 3.4.21,3.5.4, Exam June 2010, Problem 2.

Exercise 5 (week 7) Barden-Thomas: Chap. 1, Ex.1.7 (see bardenexercisechap1.pdf),

Dundas: 6.4.14, 6.4.22. (Note. One important way to show a certain subset is a submanifold is to express the subset as a level surface of a function, for a given regular value of the function; see Theorem 6.4.3 in [Dundas] or Corollary 1.3.7 in [Barden-Thomas]).

Exercise 6 (week 8) Dundas : Ex.6.3.3, 6.4.9, 6.4.24. Barden-Thomas: Ex 1.5 (with n=3).

Exercise 7 (week 9) : Barden-Thomas: Ex 2.3(see also the explanantion p.191). Dundas: 6. 4.24 (given earlier), 6.4.26.

Exercise 8 (week 10) : Barden-Thomas: Ex. 2.2. Dundas: Ex. 5.5.8 and 5.5.16 (these two exercises are essentially equivalent. Why?). Try also Ex.5.5.9 (Note that this exercise gives the solution of 5.5.8 as a special case, since the 3-sphere is diffeomorphic to SU(2)).

Here is an important hint, given as an extra exercise : Show the tangent bundle TM of an m-dim manifold M is trivial (that is, isomorphic to a product bundle) if and only if there exist m vector fields on M which are linearly independent at each point.

Exercise 9 (week 11) : Problem 1. Calculate the Riemannian metric of the Euclidean plane in terms of polar coordinates. Problem 2. Examen 2008, No. 5. Problem 3. Examen 2009, No. 4.

Exercise 10 (week 12) : Problem 1. Dundas Ex. 6.4.12; Problem 2. Dundas, Ex.9.1.4 (read first Definition 9.1.1); Problem 3. Dundas Ex. 9.2.5.

Exercise 11 (week 13) : Study the following exercise11_tma4190.pdf

Exercise 12 (week 14) :See tma4190exercise12.pdf

Exercise 13 (week 15) : See Exam June 2010, remaining problems.