In this course, we will define basic concepts of Riemannian Geometry such as Riemannian metrics, connections, geodesics, curvature, completeness, exponential map. Using this, we will study the relationships between between local and global properties of Riemannian Manifolds (e.g. Hopf-Rinow, Hadamard Theorems). Later in the course, we will specialize to study Riemannian symmetric spaces and their isometry groups.
OLGA TAUSSKY AND JOHN TODD INSTRUCTOR IN MATHEMATICS
There will be three homeworks that will be due (roughly) every two weeks, starting from the third week. Collaboration is allowed for the homework but you have to write your own solution. In the last three weeks of class, you have to make a 45 minute presentation on a topic in Riemannian geometry.
The material in this class will be taken from the following books.
1. Riemannian Geometry, Manfredo Do Carmo, ISBN: 978-0817634902.
2. Differential Geometry, Lie Groups, and Symmetric Spaces, Sigurdur Helgason, ISBN: 978-0821828489.
3. Introduction to Smooth Manifolds, John M. Lee, ISBN: 978-0387954486.