# Differential Geometry II, WiSe 2024/25

Prof. Dr. Joachim Lohkamp

Entry in the course catalog: Lecture / Tutorial; Learnweb

**Tuesday 16:15–17:45 and Thursday 12:15–13:45 in M4**

This course introduces to some of the central themes of modern Differential Geometry.

We start with the important model case of surfaces and their particularly nice curvature geometry. After a short trip through Complex Analysis, we encounter the Riemann Uniformization Theorem and the Gauß–Bonnet Theorem.

Then we turn to the higher dimensional case and to more general concepts of curvature, where analogues of the two-dimensional results may require significant generalizations or may be completely wrong. In particular, we consider uniformizations of Gromov hyperbolic spaces and the basic theory of scalar and Ricci curvature (generalizing two-dimensional results), and the Mostow Rigidity Theorem (pointing out where higher-dimensional hyperbolic manifolds behave fundamentally different from the 2-dimensional case).

We will also point out connections of these topics to other mathematical subjects like Topology, Group Theory or Algebraic Geometry, but also to General Relativity.

Prerequisites are a basic knowledge of manifolds and differential geometry as taught in the course “Differential Geometry I”.

Please enroll in the Learnweb course.

## Tutorial

Dr. Matthias Kemper

**Wednesday 16:15–17:45 in SR4**

Weekly exercises will be uploaded in Learnweb.