In this class we will study Riemannian and Differential Geometry. The official syllabus for the class is at:

Differential Geometry I

Specifically we will cover basic topics form Riemannian Geometry such as connections, geodesics, Jacobi fields, and various types of curvature. We will also study more advanced topics showing the interaction between topology and geometry. Specifically we will discuss comparison theorems leading the the famous Sphere Theorem (characterizing a sphere in terms of curvature information), harmonic forms and Bochner techniques (relating curvature and differential operators) and possibly other curvature comparison, spectral and dynamical results. I also hope to touch on other types of differential geometry such as symplectic, compelx and Kaehler geometry.

The prerequisites for the class are technically Math 6458 - Introduction to Geometry and Topology II, but an understanding of basic manifolds theory and some undergraduate analysis and algebra should be sufficient. I will start the course with a brief review of some of the main parts of manifold theory that we will need, but if you are not already comfortable with this material you will definitely want to do some reading outside of class to get caught up.

Lectures: Tuesday and Thursday 9:30 - 11:00 Skiles 271.
Professor: John Etnyre
Office: Skiles 106
Phone: 404.385.6760
e-mail: etnyre "at" math .gatech.edu

Grading Policy

Grades will be based on class attendance and either a 50 minute presentation in the Graduate Student Geometry-Topology Seminar or a 5 to 10 page paper. The talk or paper should be expository and somewhat related to differential geometry. You are welcome to choose the topic on your own or stop by my office to discuss the possibilities.

If you attend most classes and your presentation or paper is reasonable you will get an A in the class. (I recommend giving your presentation or turning in your paper early so that if I have concerns you can address them before the end of the semester.)

I will also "assign" problems during class and might post some on the class website too. These are not to be turned in for a grade, but if you really want to get comfortable with the material in the class I highly recommend trying all the problems and talking to me if you have any questions.

Textbook

I will not follow a textbook in this class, but much of the material can be found in the following two excellent books:

• Riemannian Geometry by Manfredo P. do Carmo.
• Riemannian Manifolds: An Introduction to Curvature by John M. Lee.

I will supplement the standard treatment of Riemannian Geometry with various topics. A good source for most of these are

• Riemannian Geometry by Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine.
• Riemannian Geometry and Geometric Analysis by Jürgen Jost.
• Riemannian Geometry by Peter Petersen.
• Riemannian Geometry by Takashi Sakai.