Courses

Riemannian Geometry

Course number: 250081 (Fall Term 29025/26)
Hours/ECTS credits: 2/3
Time and Place: Wed. 13:15-14:45 SR. 10 (OMP 1)
Start: 1.10.2025

General: Riemannian geometry is the study of smooth manifolds which carry a Riemannian metric, i.e. a scalar product on each tangent space. This allows to define local notions of angle, length, volume and curvature and hence to transfer the bulk of the classical elementary differential geometry of surfaces into the setting of abstract manifolds. In particular, global properties of the manifold can be studied by integrating the local contributions.

Riemannian geometry has its birthplace in Bernhard Riemann's habilitation lecture Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypotheses on which Geometry is based) of 1854. Especially since the second half of the 20th century it developped into one major branch of differential geometry with strong ties to group and representation theory as well as analysis and algebraic and differential topology. Finally it also lies at the mathematical foundations of Albert Einstein's General Relativity.

Aim and Contents: This is a first course on Riemannian geometry and provides a general introduction into the field. The natural major topics are

  • (Semi-)Riemannian metrics and manifolds
  • The Levi-civita connection
  • Geodescis, the exponential map and convexity
  • Arclength and Riemannian distance
  • The Hopf Rinow theorem
  • Curvature
  • The Einstein equations
Prerequisite for this course is a solid knowledge of manifolds and tensors as is available e.g. form the course Analysis on manifolds.

Traget audience: This lecture course primarily addresses master students specialising in geometry but is especially open to students of theoretical physics with a twist towards general relativity.

Position in the curriculum: The course is recognised as ML2 Specialisation Module or as MEL Electives in the Master Mathematik (Version 2025) respectively it is part of the core module (alternative Pflichtmodulgruppe) "Differential geometry" (code: MGED) in the area of specialisation "Geometry and Topology” in the Master Mathematik (Version 2016).

Literature, course material: This course is strongly based on the first three chapters of the standard text book "Semi-Riemannnian Geometry (With Applications to Relativity)" by Barrett O'Neill (Volume 103 of Pure and Applied Mathematics, Academic Press, San Diego, 1983).

We will follow the lecture notes Riemannian Geometry by Michael Kunzinger and myself.

Further literature which I find particularly useful includes
  • J.M. Lee, Introduction to Riemannian Manifolds (2nd edition, Springer, Cham, 2018)
  • M. do Carmo, Riemannian Geometry (Birkhäuser, Basel, 1992)
  • S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry (3rd edition, Springer, Berlin, 2004)
  • the books of Brikel Clarke, Boothby, and Kühnel as well as the lecture notes of Andreas Kriegl.

Exams will exclusively be oral and by personal appointment starting from end Jannuary. Please drop me an email approxiamtely 2 weeks prior to the desired date. The duartion of the exam is approximately 45 minutes.