# Riemannian Geometry, 1MA196, fall 2017

Main course literature:
Jost, Riemannian geometry and geometric analysis, Berlin, Springer-Verlag, 2011.

Some problems: here. (We will discuss selected ones among these in the problem sessions; others are part of your homework assignments.)

Lectures: here. (With preliminary versions of all forthcoming lectures.)

Plan of lectures:
1. Manifolds (Jost Ch. 1.1)
2. Tangent spaces and the tangent bundle (Jost Ch. 1.2)
3. Riemannian manifolds (Jost Ch. 1.4)
4. Geodesics (Jost Ch. 1.4)
► Problem session 1. (Problems 1-23 are relevant after lectures 1-4. Please digest the content of all these problems! If you are new to the topic, perhaps a good start could be to have a try on problems 6,8,12,13,17,18,23, besides the problems which are part of your homework assignment.)

5. Geodesics: Hopf-Rinow etc. (Jost Ch. 1.5 and 1.7)
6. The fundamental group. The theorem of Seifert-van Kampen.
7. Vector bundles. (Jost Ch. 2.1)
8. Vector bundles. Exterior calculus. (Jost Ch. 2.1)
► Problem session 2. (Problems 24-47 and 64 are relevant after lectures 5-8 - since I only got through half of lecture 8. In the problem session I will probably focus on (some of the) problems 26, 32*, 35, 37*, 43*, 44(a), 47.)

9. Connections (parallel transport, covariant derivative). (Jost Ch. 4.1)
10. (16 Oct) Connections (parallel transport, covariant derivative). II. (Jost Ch. 4.1)
11. (19 Oct) Curvature. Metric connections. (Jost Ch. 4.1 and Ch. 4.2)
► (3 Nov); Problem session 3. (Problems 48-68 are relevant after lectures 8.5-11. In the problem session I may focus in particular on Problems 49-51, 57, 60-61.)

12. (7 Nov) The Yang-Mills functional. (Jost Ch. 4.2)
13. (10 Nov) Levi-Civita connections. (Jost Ch. 4.3)
14. (17 Nov) Curvature of Riemannian manifolds. (Jost Ch. 4.3)
15. (20 Nov) Curvature of Riemannian manifolds. II. (Jost Ch. 4.3)
► (23 Nov); Problem session 4. (Problems 69-77 are relevant after lectures 12-15.)

16. (27 Nov) First and second variations of arc length. (Jost Ch. 5.1)
17. (1 Dec) Jacobi fields. (Jost Ch. 5.2)
18. (4 Dec) Conjugate points. (Jost Ch. 5.3)
19. (6 Dec) Comparison theorems. (Jost Ch. 5.5)
► (14 Dec); Problem session 5

Examination: There will be four compulsory assignments during the course, and an oral examination at the end of the course. Each homework assignment can give 20 points; thus the total score is 80 points. As a guideline for the final grade, a total score on the assignments above 64 will typically result in the grade 5; a total score between 50 and 63 in the grade 4, a total score between 36 and 49 in the grade 3, and a total score below 36 would result in Fail. However the final oral exam will of course also play a role for the grade, especially if the total score on the assignments is near a limit point.

Assignment #1 (due 10 o'clock on Oct 5): Problems 10 a,b   (5pt).     13 d   (5pt).     25   (5pt).     30 a,b   (5pt).

Assignment #2 (due 15 o'clock on Nov 7): Problems 46   (4p).     63   (5p).     67   (6p).     68   (5p).

Assignment #3 (due 10 o'clock on Nov 27): Problems 79   (4p).     80   (5p).     81   (6p).     82   (5p).

Assignment #4 (due 12 o'clock on Dec 18): Problems 91   (7p).     92   (6p).     93   (7p).

Oral exam: These will take place on January 10 and 11, in my office. I plan that each oral exam should take around 30 minutes, or less. Please email me to book a starting time for your exam, among the following options:
Wednesday, January 10: 9.00 (taken), 9.40 (FREE), 10.20 (taken), 11.00 (FREE), 13.00 (FREE), 13.40 (FREE), 14.20 (taken), 15.00 (taken).
Thursday, January 11: 9.00 (FREE), 9.40 (FREE), 10.20 (taken), 11.00 (taken), 13.00 (taken), 13.40 (taken), 14.20 (taken), 15.00 (taken).

(In exceptional cases I might agree to schedule the oral exam also on another time.)

Andreas Strömbergsson, Tel. (018) 4713221, e-mail: astrombe@math.uu.se