NMAK19007U Riemannian Geometry and General Relativity

Volume 2019/2020

MSc Programme in Mathematics


1. Differentiable manifolds and vector bundles.

2. Linear connections and curvature tensor

3. Riemannian metric, the Levi-Civita connection and the Einsten equation

4. Causal structure, causality conditions and global hyperbolicity

5. Extremal properties of geodesics

6. Singularity theorems or gravitational waves

Learning Outcome

At the end of the course the students are expected to have acquired the following knowledge and associated tool box:

  • the mathematical framework of (semi-)Riemannian geometry, including the Levi-Civita connection and the Riemann curvature
  • the mathematical formulation of General Relativity, including causal structure, global hyperbolicity and the Einstein equation
  • fundamental properties of solutions to Einstein's equations: singularity theorems or wave properties.



  • be able to work rigorously with problems from Riemannian geometry
  • be able to work rigorously with problems from General Relativity
  • be able to use extremal properties of geodesics to analyse singularity properties of solutions to Einstein's equation or else characterise and possibly establish existence of wave type solutions


Competences: The course aims at training the students in representing, modelling and handling physical problems of general relativity by mathematical concepts and techniques.

Lecture notes and possibly some selected articles.

Geometri 2 or corresponding knowledge of differentiable manifolds

Academic qualifications equivalent to a BSc degree is recommended.
Lectures and tutorials:

3+2 lectures (including possible seminars by students) and 2+2 tutorials per week during 8 weeks.
  • Category
  • Hours
  • Exam
  • 28
  • Lectures
  • 40
  • Preparation
  • 106
  • Theory exercises
  • 32
  • Total
  • 206
Continuous feedback during the course of the semester
7,5 ECTS
Type of assessment
Continuous assessment
The grade will be based on two major assignments to be completed in weeks 5 and 9, respectively. In addition, one must either hand in 5 smaller assignments in weeks 2,3,4,6,7 or give a seminar talk of 45 minutes about a topic to be specified during the course. If one or more of the minor assignments have not been accepted (i.e. are at least 50% correct) or the seminar talk is not deemed acceptable, the student will receive the grade -3.
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner

Half an hour oral exam without any preparation time.

Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.