Instructor: Prof. Spiro Karigiannis
Email: karigiannis@uwaterloo.ca
Registration Deadline: September 13th, 2022
Lecture Times: Monday and Wednesday | 10:30 - 11:50 am (ET)
Office Hours: Friday 10:30 - 11:30 am (ET)
Course Dates: September 7th - December 5th, 2022
Mid-Semester Break: October 10th - 14th, 2022
Registration Fee: PSU Students - Free | Other Students - $500 CAD
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.

Evaluation: Course marks will be determined as follows.

• Assignments: 100% (five assignments, roughly one every two weeks starting in third week, worth 20% each). 

Please note that you are encouraged to work together with your classmates on the assignment problems, but you must write up and turn in your own solutions to the problems.

Capacity Limit: 30 students.

Format: Hybrid.

• In-Person - MC 5479 at the University of Waterloo
• Online - Zoom

I will be lecturing in front of a blackboard to a live class of Waterloo students, and it will be simultaneously live-streamed on Zoom. Virtual participants will be able to ask questions.

Course Description

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

Textbook

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. MR1138207

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. MR2088027
• J. Jost, Riemannian geometry and geometric analysis, seventh edition, Universitext, Springer, Cham, 2017. MR3726907
• J. M. Lee, Introduction to Riemannian manifolds, second edition, Graduate Texts in Mathematics, 176, Springer, Cham, 2018. MR3887684

Brief Outline of Course topics (tentative and definitely subject to change.)

1. Review of the basics of Riemannian geometry: metrics, Levi-Civita connection, geodesics, curvature.
2. Minimizing properties of geodesics; totally normal neighbourhoods.
3. Jacobi fields and conjugate points.
4. Isometric immersions and the second fundamental form.
5. Completeness and the Hopf-Rinow Theorem; the Hadamard theorem; spaces of constant curvature.
6. First and second variations of energy; the Bonnet-Myers Theorem; the Synge-Weinstein Theorem.
7. The Rauch Comparison Theorem; the index lemma; focal points.
8. The Morse Index Theorem.
9. Existence of closed geodesics; Preissman's Theorem.
10. Cut points, the cut locus, and the injectivity radius; the Sphere Theorem.

Full Course Outline

You can find all lecture recordings here: https://www.youtube.com/playlist?list=PLArBKNfJxuunOXcRxEq1lbJ2AmwaGCo7E