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.