Math 6620, Riemannian Geometry


"All the way with Gauss-Bonnet" illustration from Spivak's book



Problem sets:
Homework 0
Homework 1
Future problem sets will be posted on Canvas.

Syllabus and course information:

Course Description.


This is a standard rigorous course in Riemannian geometry, covering Riemanninan metrics, connections and parallel transport, curvature, geodesic submanifolds, comparison theorems. We will roughly follow the topics in Lee's book "Introduction to Riemannian Manifolds".
Riemannian geometry often involves computations in coordinates, my goal will be to bring out the big picture and geometric/topological ideas, and a bit of history, so that we do not become lost in a sea of Christoffel symbols and tensors.

References.


I recommend you read along in Lee (available free from the Cornell library), we will use his notation and use this book as a roadmap. For inspiration, I will draw also from Berger's "A Panoramic View of Riemannian Geometry" and Spivak's "Comprehensive introduction to Differential Geometry" (volume II). (which you don't need to read, but they are much more inspirational, also, less rigorous). There are many other excellent texts available as well.

Prerequisites


Familiarity with smooth manifolds and differential forms is essential here. This class should be interesting and rewarding for students with a range of backgrounds, whether this is your first encounter with Riemannian geometry, or whether you have seen it in an undergrad course, or encountered a few aspects in your research.

Homework


I will assign a small number of quick problems (to make sure you are keeping up with basic definitions and notation) will be assigned each week, these are obligatory and due on Monday at the beginning of class.
I will also include, along with these, a number of thought-provoking, fun and/or more challenging problems. My goal is to frame problems that will make you curious, the kind of things that can hook your interest and perhaps even annoy you all week long, creeping up from the back of your mind while you're working on other things... You should pick at least one per week and think about it. I will be delighted to hear any progress or solutions in office hours or during/after class, and/or you can turn in written solutions at any time.

Test?


We will have a class discussion on day 1 if you would like to have an in-class test at some point during or at the end of the semester.

Final project


At the end of the semester, you will complete a ~4-5 page written project and ~20 minute presentation on a topic of your choice related to or extending the course content. Details will be given later on in the semester.

In-class work


In many class periods, you will have some allotted time to work through an example or start on a problem from the homework in small groups. Some weeks we will have volunteers present parts of homework problems at the board (e.g. if there was an interesting optional problem...)

Expectations


I expect you to attend and engage in all classes (unless of course you are ill, at a conference, etc). I will work hard to make this material interesting and get you curious about Riemannian geometry. Your end of the deal is to show up, keep up with the basic homework, and be willing to get curious about something!