Messages - Page 3

Today I finished chapter 4 and just started chapter 5.  Chapter 3 studied surfaces topologically and in chapter 4 we gave examples on how to put the simplest and most uniform geometric structure on surfaces.  Chapter 5 contains the most general theory.  It builds heavily on the tools of differential calculus, extended to abstract surfaces.  The basic constructions needed for this extension are contained in section 5.1, and it is important that you make an effort to understand this from the beginning.

A small remark on the mandatory problem set:  due to a typo there is no problem 5.

This week I have discussed classification of compact, connected surfaces.  I got as far as to show that they all can be obtained as connected sums of tori and projective planes, but it remains to study how (non-)unique this decomposition is.  We will finish this next time, and then go on to put more structure on surfaces.

Important message:The mandatory assignment has now been posted - see link on  the right side of the home page. A link to further information and instructions is also to be found there. The deadline is Thursday October 30, at 2:30 PM.

Problems for Tuesday 21:  3.1: 1, 2, 3

We have now covered all of chapter 2, except the appendix on the Beltrami-Klein model, which will not be part of the curriculum.  During the rest of the semester we will see how geometric ideas can be extended to more general situations by defining structures locally.  The first goal will be to understand the objects on which we can put such structures - topological surfaces.  We started this today by giving the definition and some examples.

Exercises for Tuesday 14:  2.8: 4, 5, 6 and 2.9: 4, 6, 7.

Note:  The mandatory exercises will be posted Thursday October 16 and must be handed in by Thursday October 30.
 

We finished 2.7 and the formulas for distance in both models.  This is used in 2.8 to generalize arc length and area formulas to the hyperbolic plane.  We did everything except the discussion of area of triangles, which will be covered next time.

Problems for Tuesday:  2.7:  5, 6 and 2.8: 2.