Lecture Course "Lorentzian Geometry"

Lecturer: Onirban Islam

Lorentzian geometry originated from the quest to understand gravity. In general relativity, gravity is the curvature of spacetime which is mathematically modelled by a 4-dimensional connected, orientable, and time-orientable smooth Lorentzian manifold. Lorentzian geometry is a richer generalisation of Riemannian geometry in the sense that all the information of a Riemannian manifold is encoded in an ultrastatic manifold but there is no converse route in general. For instance, the Hopf-Rinow theorem is impossible in the Lorentzian world albeit globally hyperbolic manifolds resemble complete Riemannian manifolds in a certain sense. The causal structure is arguably the distinctive signature of Lorentzian geometry, which has no Riemannian analogue, whatsoever. This course aims to be an introduction to the topic based on an English translation of an updated version of the lecture notes by Christian Bär.

The course will commence by introducing Lorentzian manifolds and important examples pertinent to theoretical physics, assuming rudimentary knowledge of differential geometry. Interested participants lacking this requirement will have an opportunity to familiarise themselves with some basics of smooth differential manifolds in the first tutorial which can serve as a brush-up for experienced participants. We will then treat the causality theory --- the heart of this subject and its consequences, especially Hawking-Penrose singularity theorems. Particular emphasis will be given to the modern development of Geroch's splitting theorem for globally hyperbolic manifolds due to Bernal and Sanchez.

Further information and enrolment:
All the information is available in PULS.

When and where:
Monday 08:15 - 09:45, Room 2.09. 0.13
Thursday 14:15 - 15:45, Room 2.09. 0.13
The first lecture is on 17 April 2023

Tutorial class:
Monday 10:15 - 11:45, Room 2.09. 0.14 by Rubens Longhi.

Moodle link:
Lecture notes and exercise sheets are now available in this Moodle. Participants with no account at University of Potsdam need to create a Moodle account for external students on this page.

Semester (recommended):
5. Semester or higher

Module numbers:

Necessary prerequesites:

  • Basic differential geometry (manifolds, connections, curvature, metrics)
  • Linear Algebra
  • Analysis