Universität Wien
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250070 VO Riemannian geometry (2022W)

3.00 ECTS (2.00 SWS), SPL 25 - Mathematik
MIXED
Moodle

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 05.10. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 12.10. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 19.10. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 09.11. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 16.11. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 23.11. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 30.11. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 07.12. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 14.12. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 11.01. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 18.01. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 25.01. 08:00 - 09:30 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß

Information

Aims, contents and method of the course

Riemannian geometry is the study of smooth manifolds that carry a Riemannian metric, i.e. a scalar product on each tangent space. This allows one to define local notions of angle, length, volume and curvature and hence to transfer the bulk of 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 supplies the mathematical foundations of Albert Einstein's theory of 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

Assessment and permitted materials

Oral exam

Minimum requirements and assessment criteria

Examination topics

Content of the lecture notes

Reading list

The lecture notes of the course are available here:
https://www.mat.univie.ac.at/~stein/teaching/skripten/rg.pdf
Further sources are given in these notes.

Association in the course directory

MGED

Last modified: Tu 11.06.2024 00:17