Differential Geometry II
In the lecture, basic topics in Riemannian Geometry are treated from a theoretical point of view and illustrated with several examples.
Content: A digest of the following topics will be presented:
- Exponential map and Hopf-Rinow theorem
- Connections between curvature und topology (e.g. Myers theorem, Hadamard-Cartan theorem, Klingenberg theorem, rigidity theorems)
- Closed geodesics
- Stokes theorem, cohomology
- Spaces of constant curvature, Lie groups, symmetric and homogeneous spaces
- Conformal geometry, geometric evolution equations and differential equations from geometric analysis
- Basic concepts from differential topology
(19214301)
To achieve regular and active participation it is necessary to visit at least 75% of the offered tutorials and to earn at least 60% of the possible points on the exercise sheets.| Type | Lecture with exercise session |
|---|---|
| Instructor | Prof. Dr. Konrad Polthier |
| Language | English |
| Credit Points | 10 |
| Start | Apr 16, 2024 | 12:00 PM |
| end | Jul 18, 2024 | 02:00 PM |
| Time |
|
| Note | Precondition: Differential Geometry I |
Literature
- Wolfgang Kühnel - Differentialgeometrie (English version: Differential Geometry)
- Barrett O'Neill - Semi-Riemannian Geometry
- Peter Petersen - Riemannian Geometry
-
Georg Glaeser, Konrad Polthier - Bilder der Mathematik
Exercise Sheets
- Sheet 01
- Sheet 02
- Sheet 03
- Sheet 04
- Sheet 05
- Sheet 06
- Sheet 07
- Sheet 08
- Sheet 09
- Sheet 10 (Bonus)
- Exam Preparation Sheet (with a correction from 24.9.2024)
Notes
- Lecture 01 (Manifolds)
- Lecture 02 (Vector Fields)
- Lecture 03+04 (Metric)
- Lecture 04+05 (Connections)
- Lecture 06 (Geodesics)
- Lecture 07 (Tensors)
- Lecture 08 (Curvature Tensor)
- Lecture 09-10 (Jacobi-Fields)
- Lecture 11+13+14 (Sectional Curvature)
- Lecture 12 (Locally symmetric)
- Lecture 15 (First Fundamental Group)
- Lecture 16+17+18 (Covering Spaces)
- Lecture 18+19 (Symmetric Spaces)
- Lecture 19+20+21 (Lie Groups)
- Script (Adjoint Representation)