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The lectures are now over, and the syllabus for this semester is then as follows:
Manfredo do Carmo, Riemannian Geometry:
Chapter 1 - 7, all.
Chapter 81 - 8.4.
Chapter 9.1 - remark 9.3.4.
Chapter 12, all. Here is a note on how to avoid Rauch's theorem in the proof of Lemma 12.3.1.
The exam will be June 17.
Please note that there will be no lecture this week (May 12 and 14.). Next lecture is then Tuesday May 19, when we start on chapter 12.
Note that the exam will be on June 17.
Exercises for Thursday May 6: Ch 9: 1, 2, 4, 5.
The problem set has now been posted, and you will find them at http://folk.uio.no/bjoernj/kurs/4590/obl15.pdf
Deadline is 1430 Thursday April 23.
The mandatory exercises will be posted Thursday April 9. The deadline for handing in solutions will then be Thursday April 23.
Thursday March 26 we will finish chapter 6 and do the exercises 6.1, 6.2, 6.5. This will mark the end of the local theory. After the Easter vacation we start discussing global problems, like "when do we have infinitely long geodesics?"
This week we finished the introductory discussion of curvature (ch. 4) and started on Jacobi fields (ch. 5). This is fundamental for the understanding of the geometric relationship between geodesics and curvature - hence for the rest of the course.
Exercises for next week: ch 5: 1, 2, 6, 7
This week we finished chapter 2 and started started chapter 3, defining and proving the existence of geodesics locally.
Exercises for Tuesday Feb 17: Ch 2: 1, 2, 3, 7
Note: There will be no lecture Thursday Feb 19.
I finished chapter 1 and started chapter 2, which I hope to finish next week. These chapters contain the definitions of the structure and basic tools that the rest of the course will depend on.
As problems for Tuesday I suggested Problem 4 from chapter 1 and 8 from chapter 2, but then I overlooked that 2.8 concerns concepts (Riemannian connection) that I haven't covered yet. Hence we postpone that until later.
We are now done with the short overview of the theory smooth manifolds we will need, covering basically chapter 0 in do Carmo's book. Next week we start by introducing Riemannian metrics.
Exercises from chapter 0: 2, 4, 5, 11 (for Tuesday Feb,3.)
This course will this semester meet twice a week: Tuesdays 1215-1400 and Thursdays 1015-1200, both times in room B738. The next lecture will then be Thursday Jan.22.
In the first lecture I just discussed the basic definitions of smooth manifolds, tangent vectors and derivatives. A leisurely introduction to this material, following my notation, can be found in section 5.1 of my notes for the course MAT4510 Geometric structures.
Somehow nobody had noticed an unfortunate collision in the time schedules of MAT4520 and MAT4590. One possible solution is for MA4590 to meet only two hours on Tuesdays (1215-1400) and then again for two hours Thursdays 1015-1200. This way we will also have time for at least one hour of Exercises every week.
We will meet as planned Tuesday January 20 and discuss this further. It is therefore important that you come, if you plan to follow the lectures this term.
We will use the book Riemannian Geometry, by Manfredo Do Carmo, and the plan is to go through most of chapters 1-9 and chapter 12. The subject uses some of the language and concepts from the theory of differential manifolds, but it suffices to take MAT4520 at the same time. Much of this is also a straightforward generalization of theory from MAT4510 - Geometric structures.