This Fall, I am teaching a graduate course on Einstein manifolds.

In this course we will study some topics in Riemannian and pseudo-Riemannian geometry. We will mostly focus on Ricci curvature and its applications. The course will start with basics about Riemannian and pseudo-Riemannian geometry. We will assume familiarity with differential manifolds and basic calculus on them.

We will cover the following topics:

Linear connections on vector bundles: Torsion, Curvature, Bianchi identities
Riemannian and pseudo-Riemannian manifolds
Get the feel of Ricci curvature: Volume comparison theorems, Bonnet-Myers theorem
Ricci curvature as a PDE
Einstein manifolds and topology
Homogeneous Riemannian manifolds
Kahler and Calabi-Yau manifolds
Quaternion-Kahler manifolds

The main reference for the class will be: A.L. Besse: Einstein manifolds, Springer, 1987.

Due to the Covid pandemic those lectures are online and the videos are publically posted on a dedicated webpage.