Spiro Karigiannis: A Second Course in Riemannian Geometry

This is a 24-lecture course, with each lecture being about 80 minutes, given by Spiro Karigiannis.

This is a second course in Riemannian geometry. The emphasis will be on the intimate relationship between curvature and geodesics.

Prerequisites: Students should be thoroughly familiar with smooth manifold theory, and some exposure to the basics of Riemannian geometry, including Riemannian metrics, the Levi-Civita connection, Riemann curvature, and Riemannian geodesics is helpful but not absolutely essential.

Textbooks: There is no required textbook for this course. I will be following this book quite closely, however:

• M. P. do Carmo, Riemannian geometry, translated from the second Portuguese edition by Francis Flaherty, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992.

I will likely change notation from do Carmo, and I will certainly change the sign and normalization conventions for curvature to the standard ones. Other useful references are:

• S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry, third edition, Universitext, Springer-Verlag, Berlin, 2004.
• J. Jost, Riemannian geometry and geometric analysis, seventh edition, Universitext, Springer, Cham, 2017.
• J. M. Lee, Introduction to Riemannian manifolds, second edition, Graduate Texts in Mathematics, 176, Springer, Cham, 2018.

1. Lecture 1
2. Lecture 2
3. Lecture 3
4. Lecture 4
5. Lecture 5
6. Lecture 6
7. Lecture 7
8. Lecture 8
9. Lecture 9
10. Lecture 10
11. Lecture 11
12. Lecture 12
13. Lecture 13
14. Lecture 14
15. Lecture 15
16. Lecture 16
17. Lecture 17
18. Lecture 18
19. Lecture 19
20. Lecture 20
21. Lecture 21
22. Lecture 22
23. Lecture 23
24. Lecture 24

These videos were produced by the Fields Institute, as a graduate course (link to course page).