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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.
TypeLecture with exercise session
InstructorProf. Dr. Konrad Polthier, Dr. Tillmann Kleiner
LanguageEnglish
Credit Points10
StartApr 15, 2026 | 10:00 AM
endJul 18, 2026 | 12:00 PM
Time
  • Lectures: Monday 10:00-12:00 in A6/SR 025/026 (Arnimallee 6) and Wednesday 10:00-12:00 in A6/SR007/008; (starting April, 15th, 2026 - Wednesday in the first week of the semester)
  • Tutorial 1: Monday 8:30-10:00; A6/SR 025/026 (Arnimallee 6); (starting April 20th, 2026)
  • Tutorial 2: Tuesday 8:15-9:45; A6/SR 032 (Arnimallee 6); (starting April 21st, 2026)
  • Exam 1: Written exam of 90 minutes in the time frame 10-12am on Juli 27th, 2026 (Monday) at HFB/D (Henry-Ford-Bau, Lecture Hall D, Garystr. 35)
  • Exam 2: Written exam of 90 minutes in the time frame 10-12am on September 28th, 2026 (Monday) at HFB/C (Henry-Ford-Bau, Lecture Hall C, Garystr. 35)
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:

Lecture Notes:

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