Math 6240 (Fall 2020):
Differential Geometry 2:
Riemannian Geometry

[homework] [lecture notes] [lecture video] [feedback]

Meetings

Class meets remotely at 11:30am on MWF in Zoom Meeting Room 93961663776. If you are enrolled in the class you will get an e-mail with the password before the first class. If you are not officially enrolled but want to attend, please e-mail me to get the password.

This class is officially listed as a "hybrid" class, which means that there is a classroom (ECON 13) reserved for our meetings. We will not meet in this classroom, but you may wish to use the room if you need a place to watch our lectures online. If you do use this room, please follow all of the required safety measures.

Office hours will also be held remotely in the same Zoom Meeting Room 93961663776, either by appointment or according to the schedule below.

For at least the first few weeks of the semester, weekly attendance of office hours will be required.

Textbook

The textbooks will be Introduction to Riemannian Manifolds, by John M. Lee, and Differential Geometry: Connections, Curvature, and Characteristic Classes, by Loring W. Tu. We will also refer back to Lee's Introduction to Smooth Manifolds and. We will use Foundations of Differentiable Manifolds and Lie Groups by Frank W. Warner for our discussion of sheaves and the de Rham theorem. All are available by the University of Colorado's subscription to SpringerLink. To access the books, make sure you are on the University of Colorado network, either by being physically on campus, logging in through the library, or using some other form of VPN.

Syllabus

  1. Review of differential geometry
  2. Vector bundles and sheaves
  3. de Rham cohomology
  4. Riemannian metrics
  5. The Gauss—Bonnet theorem (for surfaces)
  6. Connections, and curvature
  7. Levi-Civita connection
  8. Geodesics
  9. Review of Lie groups and Lie algebras
  10. Characteristic classes

Lecture video

  1. Lecture 26: first-order trivialization of metrics
  2. Lecture 25: length and volume
  3. Lecture 24: Riemannian metrics
  4. Lecture 19: Poincar&eacture; duality and Lefschetz duality
  5. Lecture 18: sheaf cohomology, soft sheaves
  6. Lecture 17: sheafification, sheaf cohomology
  7. Lecture 16: espace étalé, equivalence of two definitions of sheaves
  8. Lecture 15: sheaves as functors, espace étalé
  9. Lecture 14: compactly supported cohomology of oriented manifolds, sheaves as topological spaces
  10. Lecture 13: homological algebra, snake lemma, Mayer-Vietoris, cohomology of spheres
  11. Lecture 12: compactly supported cohomology, homotopy invariance of cohomology
  12. Lecture 11: Lie derivative, de Rham cohomology
  13. Lecture 10: exterior derivative, Lie derivative
  14. Lecture 8: tensor products of vector bundles
  15. Lecture 7: alternating multlinear functions, contractions
  16. Lecture 6: alternating multlinear functions, tensor products
  17. Lecture 5: Grassmannian, multilinear algebra
  18. Lecture 4: pullbacks of vector bundles, the Grassmannian
  19. Lecture 3: sections of vector bundles
  20. Lecture 2: vector bundles
  21. Lecture 1: the inverse function theorem

Homework

  1. Assigned Wednesday, November 25: Lee II, 4-2, 4-4, 4-9, 4-14, 5-3, 5-10, 5-11, 5-23.
  2. Assigned Tuesday, November 10: Lee II, Chapter 2 2-1, 2-3, 2-7, 2-16, 2-32, 2-22, 2-26. I'm curious to see whether any of you have a better solution to 2-26 than what I came up with.
  3. For Friday, October 23: Lee I, Chapter 18, Problems 18-7, 18-8, 18-9; Hartshorne, Exercises II.1.13, III.2.7.
  4. For Friday, October 16: Lee I, Chapter 18, Problems 18-7, 18-8; Hartshorne, Chapter II, §1, Exercises 1.2, 1.8, 1.13.
  5. For Friday, October 9: Lee I, Chapter 17, Problems 17-10; Lee I, Chapter 18, Problems 18-1, 18-2, 18-6, 18-9; Warner, Chapter 5, Exercises 3, 4, 17.
  6. For Friday, October 2: Lee I, Chapter 17, Problems 17-4, 17-6, 17-7, 17-8, 17-10, 17-12, 17-13.
  7. For Friday, September 25: Lee I, Chapter 12, Problems 12-10, 12-11, 12-12 and Lee I, Chapter 14, Problem 14-9 and Lee I, Chapter 17, Problem 17-1, 17-12. If you need a reminder about calculating with differential forms, you may want to do 14-6 and 14-7.
  8. For Friday, September 18: Lee I, Chapter 14, Problems 14-1, 14-3, 14-4, 14-5. Reminder: discuss the problems on Zulip.
  9. For Friday, September 18: Lee I, Chapter 12, 12-1 – 12-6, 12-10 – 12-12.
  10. For Friday, September 4: Lee I, 10-1, 10-2, 10-5, 10-6, 10-9, 10-13, 10-15, 10-16, 10-18; Lee II, 4-7 (compare to 10-9); Tu, 7.1.
  11. Monday, August 31: Make at least one post to Zulip about an exercise.

Reading

  1. Friday, November 20: We are going to talk about the Levi–Civita connection and a bit about curvature. The main topics will be torsion of the Levi–Civita connection (pp. 115–126), the definition of curvature (pp. 193–199), and examples (pp. 136–145) in Lee II.
  2. Tuesday, November 17: We are discussing connections and the Levi–Civita connection in particular. These are covered in Lee II, Chapters 4 and 5.
  3. Friday, November 13: Read Lee II, Chapter 4.
  4. Saturday, November 8: Read Lee II, Chapter 2 and/or Lee I, Chapter 13.
  5. Tuesday, October 27: Read Warner, Chapter 5, pp. 191 – 199.
  6. Tuesday, October 13: Read Warner, Chapter 5, pp. 173 – 185. For another introduction to sheaves, you may want to look at Hartshorne, Chapter II, §1 or Tennison, Chatpers 1–3.
  7. Saturday, September 26: Read Warner, Chapter 5, pp. 162 – 173. Please also start looking at the proof in Lee I, Chapter 18.
  8. Friday, September 18: Read Lee I, Chapter 17.
  9. Wednesday, September 9: Finish reading Lee I, Chapter 14. The next topics will be Chapters 17 and 18.
  10. Wednesday, September 2: Read Lee I, Chapter 14, pp. 349—359. This covers alternating multilinear maps, which are only covered briefly in Chapter 12.
  11. Monday, August 31: Read Lee I, Chapter 12 and/or Tu, Chapter 4. There is a lot of overlap, so you don't necessarily need to read both. I will try to cover pp. 303—316 of Lee on Wednesday and the rest on the following Monday.
  12. Monday, August 24: Read Lee I, Chapter 10 and Tu, §§7, 20.4—20.5.
  13. Before the semester: Review material from Differential Geometry 1. Read Lee II, Appendix A.

Grading

To get an A in this course, you should participate actively in class, and present homework solutions regularly, either during class discussion, in office hours, or in written form.