August 22. and 25. 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.
I then started lecturing from chapter 2, formulated the axiom H (p. 12) and started to discuss how to construct models for hyperbolic geometry. To this end, I defined stereographic procjection and derived coordinate formulas for stereographic procjection and its inverse. On Monday 29. I will proceed with section 2.1 and I will also start lecturing from section 2.2 (from the lecture notes by Jahren).
August 29. and September 1. I started lecturing from 2.1, proving Lemma 2.1.1 and 2.1.2. I then lectured from 2.2 ending with Proposition 2.2.9. I also went through Problem 2.2.5.
On Monday September 5. I completed section 2.2 and I also gave some introductory comments to section 2.3.
Thursday September 8. I looked at the assigned problems. Then I started lecturing from 2.3 (ending approximately on the middle of p.30)
Monday September 12. I completed the classification of positive real M?bius transformation. I also gave some introductory comments on the classification of the negative real M?bius transformations. On Thursday I will first look at the assigned problems and then proceed with the classification of the negative real M?bius transformations.
Monday 19. I completed the classification of negative M?bius transformations. I then started on section 2.4, defined congruence in H and formulated and proved 2.4.2, 2.4.3 and 2.4.4
Thursday 22. I looked at the problems 2.3.1, 2.3.3, 2.3.4 and 2.3.5. I then proved Lemma 2.4.5.
Monday 26. I proved that H satisfies the congruence relations C1-C6. I then started with defining a metric on H and proved that this metric satisfied the condition d1, d2, d5. I leave to you to read the proof that d3 and d4 also is satisfied.
Thursday 27. I first looked at the assigned problems. Then I lectured 2.5 and started on 2.6 ending with the normal form for transformations in M?b^+(D) (formula 2.7.1).
Monday October 3. I completed section 2.7. On Thursday I will start on 2.8.
Thursday October 6. I lectured most of 2.8
Monday October 10. I completed 2.8 and most of 2.9 ending by proving the the second law of cosine. On Thursday I will start by looking at the assign problems, then I will complete 2.9 and start lecturing from chapter 3.
Thursday October 14. I did the assigned problems. Then I completed 2.9 and gave some introductory comments about the topological classification of cmpact surfaces(chapter 3).
Monday October 17. I started lecturing from chapter 3. I did not quite manage to complete the proof of theorem 3.19. On Thursday I will start at the top of page 89, showing the argument of how to split out factors of the type T^2. On Thursday I will complete chapter 3 and also do the problems 4, 5, 6 and 7 (from this chapter).
Thursday October 21. I completed chapter 3 and did the problems 4,5, 6 and 7.
Monday October 24. I began lecturing section 4.1 on Thursday I start by explaining how to put a hyperbolic structure on the double torus (middle of page 99). I will then complete section 4.2 and begin lecturing from chapter 5.
Thursday October 27. I completed section 4.2 and strarted lecturing from chapter 5.
Monday October 31. I proceeded lecturing from chapter 5. Thursday October 3. I will start on 5.3, Riemannian surfaces.
Thursday November 3. I covered 5.3 and most of 5.4.
Monday November 7. I completed 5.4, started on 5.5 and ended(unfortunately) approximately in the middle of the proof of Theorema Egregium (p. 131).
On Thursday I will complete this proof and the remaining part of section 5.5. I will then, as examples, do the problems 5.1.4, 5.3.1 and 5.5.1. Then I will start on section 5.6.
Thursday November 10. and Monday November 14. On Thursday I completed section 5.5 and started on section 5.6, I proceeded on section 5.6 on Monday ending by deducing the geodesic equation of the hyperbolic half-plane (p. 144).
Thursday November 17. I completed section 5.6 and started on section 5.7 ending the lecture by proving Gauss? Lemma(5.7.2)
Monday November 21. I finished 5.7 and 5.8 and started on 5.9 by formulating and explaing The Gauss Bonnet Theorem. On Thursday, I will say a little bit more about the theorem and start on the proof of Gauss Bonnet.