August 23. and 24. I went through the short note about Hilbert axioms. I also gave some examples of how to apply these axioms. This stuff is merely meant as motivation for especially the hyperbolic geometry which I will start lecturing on Monday Aug. 30.
August 30. and 31. I started lecturing from the note about hyperbolic geometry, and I covered the topics up to (and including) Proposition 2.8.For next week I assigned the problems 1.2, 1.4, 2.2 and 2.5, and I will start looking at these problems next Monday, and then proceed lecturing theory about hyperbolic geometry.
September 6. and 7. On Monday I started looking at the problems I assigned last week. Then I proceeded lecturing from section 2 in the notes, and started on section 3 about classification of real M?bius transformations. I am ended the lecture starting classifying M?bius transformations with two real fix points (case 2 p. 16) and I will proceed with this next Monday. For next week I have assigned the problems 2.3, 2.6 and 2.7.
September 13. and 14. On Monday I finished the classification of real M?bius transformations in M?b^+(H). Then I solved the problems 2.3, 2.6. Tuesday I classified the M?biustransformations in M?b^-(H) and then solved problem 2.7. For next week I assigned the problems 3.1, 3.5 and 3.6.
September 20. and 21. On Monday I defined the intermediate relation in H, defined the congruence relation on the set of segments and angles. I then formulated and proved the Lemma and Corollaries 4.2-4.5 and proved the congruence axioms for segments. Tuesday I proved the remaining congruence axioms for angles and I solved the problems 3.1, 3.5 and 3.6.
September 27. and 28. On Monday I defined the hyperbolic metric in H and proved that (apart from being a metric) it is satisfies the properties given in (d4) and (d5) (p. 26). Then I defined the homeomorphism G from D to D. On Tuesday I explained what we mean by a M?bius transformations in D and gave general formulas for these.Then I defined the hyperbolic metric in D and started to calculate formulas for this metric. I will proceed with this next week, and then proceed with chapter 8. (I will talk about the stuff in chapter 6, when I deduce the area formula for an hyperbolic triangle). I assigned the problems 2.9, 5.1 and 7.8 for next week.
October 4. and 5. On Monday I completed section 7. and started on section 8 ending by explaining the area integral formula in H. I proceeded with section 8, but also talked a little bit about angle measure (section 6) before I proved the area formula for a hyperbolic triangle. I then completed section 8 and I will finish the notes on hyperbolic geometry next week and the start on the notes about classification of compact surfaces.
October 11. and 12. On Monday I talked about hyperbolic trigonometry and I finished the notes about hyperbolic geometry (I dropped the appendix). Then I started lecturing from the note about the topological classification of compact, connected surfaces. I defined the standard surfaces S^2, P^2, T^2, explained the construction of forming the connected sum of two surfaces, and I stated the classification Theorem (Theorem 1). On Tuesday I started by explaining (in a rather informal way) about orientation of surfaces. I then explained how we can regard S^2, P^2 and T^2 as quotient spaces of type D^2/W (quotients of the disk D^2 where the equivalence relation is coded into a word W) and I then gave the proof of Theorem 1. Next week I will finish the notes about the classification theorem. I will also look at most of the exercises in these notes, and perhaps start to talk about notes no. 4 ("Geometry on surfaces-basic concepts")
October 18. and 19. On Monday I finished the notes about the topological classification of connected compact surfaces. I also solved the problems 1, 2, 3 4, 5 and 7 from these notes. Tuesday I started looking at the notes " Geometry on surfaces-basic concepts" and I have explained some of the examples, defining geometrical structures on some surfaces. Next week I will complete these notes starting by looking further into the examples on p. 5 . I will then start lecturing from the notes " Differential geometry on surfaces."
October 25. and 26. On Monday I finished lecturing from the notes " Geometry on surfaces-basic concepts". I then started lecturing from notes " Differential geometry on surfaces." I have completed the two first sections (I dropped the proof of Prop. 2.2) and I will proceed with lecturing about Riemannian surfaces on Monday (section 3 of from the notes).
November 1. and 2. On Monday I repeated the definition of a Riemannian surface and I lectured the stuff in section 3 and 4 in the notes and ended by giving the definition of the Gaussian curvature of regular surfaces in R^3 (section 5). On Tuesday I proceeded lecturing from section 5 and ended the lecture by proving Theorema Egregium. Next week I will add some comments about curvature, I will then look at the problems 1.4, 3.1, 5.1 and 5.2 and then proceed by lecturing about geodesics (section 6).
November 8. and 9. On Monday I gave proof of Proposition 5.6 (in detail). I then gave the solutions of the problems 1.4, 3.1, 5.1 and 5.2. I then started lecturing about geodesics (section 6). On Tuesday I proceeded lecturing about geodesics. I completed section 6, and I started on section 7 introducing geodesic polar coordinates. Next week I will proceed with lecturing from section 7. I will then look at the problems 4.6, 5.5 and 7.2. All these problems are about what happens if scale the metric with a constant factor. We will need these results in section 8 about spaces with constant Gaussian curvature. After looking at these problems I will proceed with section 8 and perhaps start lecturing from the last section about Gauss Bonnet theorem.