Riemannian Geometry

Course content

1. Differentiable manifolds and vector bundles.

2. Linear connections and curvature tensor

3. Riemannian metric, the Levi-Civita connection

4. Curvature

5. Geodesics and the exponential map

6. Extremal properties of geodesics


MSc Programme in Mathematics

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 Riemannian geometry, including the basic theory of vector bundles
  • the Levi-Civita connection
  • the Riemann curvature tensor and its basic properties including the Bianchi identities
  • immersed submanifolds and the second fundamental form, including examples
  • geodesics and the exponential map and extremal properties



  • be able to work rigorously with problems from Riemannian geometry
  • be able to treat a class of variational problems by rigorous methods
  • be able to use extremal properties of geodesics to analyse global properties of manifolds


Competences: The course aims at training the students in representing, modelling and handling geometric problems by using advanced mathematical concepts and techniques from Riemannian geometry.

Lectures and tutorials:

3+2 lectures (including seminars by students) and 2+2 tutorials per week during 8 weeks.

Lecture notes and/or textbook

Geometri 2 or corresponding knowledge of differentiable manifolds

Academic qualifications equivalent to a BSc degree is recommended.

Continuous feedback during the course of the semester
7,5 ECTS
Type of assessment
Continuous assessment
7 written assignments during the course of which the 5 best count equally. In addition, one must give a seminar talk of 45 minutes about a topic to be specified during the course. The written work and seminar talk count with equal weights in the final grade.
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner
Criteria for exam assessment

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

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 40
  • Preparation
  • 106
  • Theory exercises
  • 32
  • Exam
  • 28
  • English
  • 206