Report from the lectures
Aug. 24. In the first two lectures I went through the first chapter of the course notes on Hilbert's axioms and did some examples to show how the axioms are used. This is just a preliminary chapter which should serve as a motivations for some of the things we will do later in the course. Next time I start defining hyperbolic geometry. As a preparation you should read the first section on stereographic projection.
Aug. 27. Today I started on the hyperbolic geometry notes. The idea is to attempt to construct a model for hyperbolic geometry on the open unit disc with lines given by chords. In section 1 we show that this is equivalent to a model based on the upper half plane with lines given by semi-circles with center on the real axis or vertical lines. Then we will see that we can define congruence as equivalence under certain fractional linear transformations. This is the topic of section 2, which I started and covered down page 7. Problems for next time: 1.2, 1.4, 2.2, 2.10 (except last question).
Aug. 31. I used too much time on the exercises today, and only finished page 9 in the notes. Exercises for next time will be 1.3 and 2.4.
Sept. 3. I finished section 2 and gave a preview of section 3. Problems for next time: 2.5 and 2.6.
Sept. 7. I started discussing classification of Möbius transformations. Problems: 3.3 and 3.5.
Sept. 10: Today we finished section 3. Problems: 3.1, 3.4, 3.7, 3.8.
Sept. 14: Today I discussed how the group of Möbius transformations can be used to define congruence in hyperbolic geometry. The technical results 4.2-4.5 are essential here, showing that Möb(H) is "just big enough" to prove existence and uniqueness in C1 and C4. In fact, Möb+(H) would have been sufficient for C1-C5, but for the most general version of C6 (the SAS congruence criterion), we need the whole group.
There are no exercises on this material, so next time I start directly on section 5, establishing the hyperbolic plane as a metric space.
Sept. 17: I went through section 5, constructing a distance function (metric) in the upper half-plane. Exercises for next time: 5.1-3.
Sept. 21: After a brief discussion of angle measure and the upper half-plane as a conformal model for the hyperbolic plane, I started on section 7. Next time we start with the formula for the metric in the Poincare disk model and go on to the more general arc-length discussion of section 8. Exercises: 6.1, 6.2, 7.1, 7.3.
Sept. 24: Today I finished chapter 7 and the arc-length part of chapter 8. Based on a discussion with the students present, I will from now on do more exercises in class and leave some of the theory to self-study. Exercises for next time: 7.5-7, 8.1-2.
Sept. 28: We finished chapter 8. Problems for next time: 8:3, 8:5, 8:6. Exam 2007:1, Exam 2008:3.
Oct. 1: Today we finished chapter 9, and hence the hyperbolic geometry part of the course. Next time we start on part 3 of the notes: "Classification of compact, closed surfaces". Exercises: 9.1, 9.4, 9.5, 9.6.
Oct. 5: I started on compact surfaces, discussed connected sum and orientability and formulated Theorem 1. There will be no exercises for next time, and I will finish the classification then.
Oct. 8: I finished the proof of the classification theorem for compact surfaces. Next time we do all the exercises in the notes, and then go on to discuss how to put geometric structures on surfaces.
Remember that the mandatory assignment will be posted Friday Oct. 9. (Around noon.)
Oct. 12: We started on the Introduction to geometric structures notes. First I discussed in general terms the necessity for coordinate transformations to preserve structure, and then used Euclidean structure on the torus as an explicit example. Next time I will discuss more examples, before starting on the last part, on differential geometry.
Note that there will be no lectures October 19 and October 22.
Oct. 15: I continued the discussion of geometries on various surfaces. Next lecture will be Monday October 26, and then we start on the last part of the course: Differential geometry on surfaces.
Oct. 26: The study of geometry on surfaces relies heavily on notions of change - i.e. derivation. The first section sets up the machinery necessary for this, and Monday's lecture was devoted to motivation and the basic idea of tangent planes as liner approximation of the surface. Thursday we continue this, and the plan then is to finish sections 1 and 2 and start section 3. Exercises: 1.3-5.
Thursday I will also give back the mandatory exercises.
Oct. 29: Today I finished the disussion of the basic tools and language of sections 1 and 2. This will be taken for granted and used constantly in the rest of the course. So will the exercises (1.3-5), which will be the basis for many examples later.
Exercises for next time: 1.1, 1.2, 2.3
Nov.2: I went through the fundamental definitions of section 3 in detail, stressing how everything can be seen as natural generalizations of concepts we already are familiar with. Section 4 is left to self study, and next time I start section 5 on curvature - perhaps the most important concept of differential geometry.
Exercises: 3.1, 3.2, 4.2, 4.4
Nov. 5: I defined and discussed the fundamental concept of Gaussian curvature on a regular surface in R³, and proved the important Proposition 5.2. Exercises for next time: 5.1, 5.3, 5.4.
Nov. 9: Today I started with a description of how regular surfaces locally can be thought of as graphs of functions. This means that the example following Proposition 5.2 is quite general. I finished section 5 with a discussion of Gauss' Theorema Egregium and its two important consequences: Gaussian curvature is invariant under local isometries, and it can be defined for abstract, Riemannian surfaces.
Exercises: 4.5, 4.6, 5.5, 5.6
Nov. 12: We started with a general discussion of what "lines" should be on abstract surfaces and went on to define and study the second covariant derivative of curves on regular surfaces in R³. The fundamental result is the local formula (17), page 23 in terms of E, F, G and their partial derivatives, showing that the second covariant derivative is intrinsic and can be defined for curves on abstract Riemannian surfaces.
Exercises for next time: 6.3, 6.4, 6.5, 6.7
Note: There will be no lecture Thursday November 19th!
Nov. 16: Today I finished section 6. I want to emphasize the importance of the exponential map and Theorem 6.4. Next time it will be used to define geodesic polar coordinates, which will be needed when we will classify metrics of constant curvature. (Section 8.)
Exercises: 6.1, 6.2, 6.6, 6.8.
Nov. 23: In section 5 we studied curvature and in section 6 we studied geodesics and defined the exponential map. In section 7 the two come together, in the formula for the exponential map in polar coordinates. One way of thinking about the formula is that it describes how curvature at a point controls how geodesics spread out from the point. Negative curvature means they spread out more, positive less, than in the Euclidean case.
In section 8 this is applied to prove that surfaces of constant curvature are locally isometric to scaled versions of the classical geometries.
The last two lectures will be devoted to the famous Gauss-Bonnet theorem. Exercises: 7.1, 7.2 and problem 3 from the final exam 2007.
Nov. 26: Today I started discussing the Gauss-Bonnet theorem, which gives a relation between differential geometric invariants (curvature) and purely topological information (Euler characteristic). This is one of the most famous theorems of geometry, and is a model for numerous generalizations and analogous results in modern mathematics. I discussed the form of the theorem and defined the terms involved, and next time I will finish the proof and discuss some applications.
Exercises: 9.4, 9.5, Exam 2006 problem 3.