Riemannian geometry and general relativity (Spring term 2023)

Content

The aim of this lecture is to introduce fundamental concepts from global geometry on manifolds, in particular, the so-called (semi-)Riemannian manifolds. Basically, ideas from linear algebra such as metrics and distances in Euclidean vector spaces are generalized to manifolds, where metrics are defined on any tangent space in a compatible (namely, differentiable) way. This gives rise to suprisingly rich geometric structres, such as curvature, covariant derivatives, geodesics etc. A particularly nice application are the mathematical foundations of general relativity. The lecture will start with a brief introduction (or reminder) of some notions from topology, such as manifolds, vector fields and their flows etc. Then we will introduce (semi-)Riemannian metrics and the various curvature notions. Applications will be given if time permits. This class can be attended with or without prior knowledge in classical differential geometry (treating essentially the geometry of curves and surfaces).

This lecture will be given in either German or English, depending on the audience.

There will be a weekly exercise sheet. Working actively on solving the problems from these sheets is an essential part of the lecture.

Prerequisits for this lecture is a good account of the content of the lectures " Analysis 1,2" and " Lineare Algebra und analytische Geometrie 1 und 2".

Literature

Helga Baum:Eine Einführung in die Differentialgeometrie, Vorlesungsskript, HU Berlin, 2006.
Barrett O'Neill: Semi-Riemannian Geometry, Academy Press 1983.
Manfredo do Carmo: Riemannian Geometry, Birkhäuser 1992.
Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine: Riemannian Geometry, Springer 1990.
Shoshichi Kobayashi, Katsumi Nomizu: Foundations of Differential Geometry I and II, Wiley and Sons, 1996.

Lecture and exercises

Please enroll at this address, where you will also find any other information related to the lecture (exercise sheet, manuscript etc.). Time slots for the lectures are:

Tuesday, 13:45–15:15, room W017 (C25.017)
Wednesday, 13:45–15:15, room N002 (C10.002)

Exercise classes (by Henry Dakin)

There will be a weakly exercise sheet related to the content of the lecture. It is highly recommended to solve the exercises in order to understand the material from the lecture properly. These exercises will be discussed in a weekly exercise class, in which you can also ask any other question related to the content of the lecture. The exercise classes will take place every Friday, 13.45-15.15, room N105 (C10.105)