# Report from the lectures

August 20. and 22. I went through chapter 1 about Hilbert´s axioms. I also gave some examples of how to apply these axioms. This stuff is merely meant as motivation for especially the hyperbolic geometry.

August 27. and 29. Started lecturing from chapter 2 in the notes. I covered the stuff including Lemma 2.2.3.

September 3. I proceeded lecturing from from 2.2. I completed this section and started on section 2.3 (classification of Möbius transformations)

September 5. I completed the classification of Möbius transformations in Möb^+(H).

September 10. As examples I classified the two first maps in problem 2.3.1. I then completed section 2.3, classifying the transformations in Möb^-(H).

September 12. I did the problems 2.2. 2, 3, 4, 6 and 7. I then started on section 2.4 defining congruence relations in H.

September 17. I repeated the definition of congruence relations, stated and proved the Lemmas 2.4.2-2.4.5 and proved that H satisfied the axioms of congruence for segments and angles.

September 19. I did problems 2.3 2, 4, 5 and 6. I then explained how we can define a metric in H adapted to the hyperbolic structure (section 2.5).

September 24. I looked at angle measure (2.6) and started on section 2.7 defining and describing Möb^(D).

September 26. I looked at the problems 2.3. 7, 8 and 9 and 2.5. 1. I proceeded with 2.7 explaining that the subgroup fixing 0 in Möb^(D) is equal O(2).

October 1. I gave more formulas for hyperbolic metric in D and in H (2.7). Then I explained how to calculate hyperbolic arc-length in H and D (2.8).

October 3. I looked at problem 2.3.10 and the problems 2.5.2.3 and 4. Then I explained how to define area in H and D (2.8).

October 8. I explained that hyperbolic area is invariant under möbius transformations, and calculated the area of hyperbolic triangles and circles. Then I started on 2.9 stating and proving the first law of cosine and the hyperbolic law of sine. I ended by stating the second law of cosine and explaining SSS and AAA in hyperbolic geometry.

October 10. I completed the section about hyperbolic trigonometry and I moved to chapter 3. I gave examples of compact surfaces talked about connected sum of such surfaces, about orientation, and I ended by stating the main Theorem in this chapter. (Theorem 3.1.9.)

October 15. I did the proof of Theorem 3.1.9. and also explained Theorem 3.1.13.

October 17. I proved the formulae for the Euler characteristic of S(m,n). I moved to chapter 4. Explained what we mean by a 2-dimensional smooth surface, somewhat loosely, what we mean by a geometrical structure on a surface. I finally gave examples of how we can put, in various ways, geometrical structure on a tori.

October 22. I completed chapter 4.

October 24. I did the assigned problems.