# Fields Academy Shared Graduate Course: Topics in Geometry and Topology: A Second Course in Riemannian Geometry

## Description

**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.**)

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