# Vorlesung: Differential Geometry II/Differentialgeometrie II (summer semester 2021)

The lectures takes place on Wednesday and Thursday 11:15–12:45.
The tutorial takes place on Thursday 13:15–14:45.
**The Zoom invite can be found on the Moodle page.**

## Prerequisites

This course assumes some familiarity with Differential Geometry. If you have taken Differential Geometry I in WS20/21, then you are more then well-prepared.## Lecture Notes

Here are my live lecture notes.## Topics

This is a course on Riemannian Geometry. I plan to cover the following topics (not necessarily in this order):- What is a Riemannian metric?
- Covariant derivatives, the Levi-Civita connection, the Fundamental Theorem of Riemannian Geometry
- Geodesics, geodesic flow, the exponential map
- Riemannian curvature; sectional, Ricci, scalar curvature
- Second variation and Jacobi fields
- Spaces of constant curvature, the Riemann–Killing–Hopf Theorem
- Notions of completeness, Hopf–Rinow Theorem
- Spaces of non-positive curvature, Hadamard's Theorem
- A first glimpse at the interaction between topology and geometry: Gauß–Bonnet theorem
- Bonnet–Myers Theorem
- The Bochner technique: Killing fields; harmonic forms; Hurwitz' automorphism theorem for Riemann surfaces
- The Lichnerowicz–Obata Theorem
- Comparison geometry; Rauch's Theorem, Bishop–Gromov, applications
- Holonomy

- Morse theory of energy functional
- Chern–Weil theory and the Chern–Gauß–Bonnet theorem
- Cheeger–Gromoll splitting theorem
- Hodge theory
- Gromov's Betti number bounds