# 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.