Riemannian Geometry

Course content

  1. Differentiable manifolds and vector bundles
  2. Linear connections and the 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
MSc Programme in Mathematics with a minor subject

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
Type of assessment details
7 written assignments during the cours 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

Resubmission of all 7 assignments from the continuous assessment of which the 5 best count equally. Deadline at 12 o'clock noon on Friday in the reexamination week. Additionally, a half an hour oral reexam without any preparation time.

The written work and oral reexam count with equal weights in the final grade.

Criteria for exam assessment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.

Single subject courses (day)

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


Course number
7,5 ECTS
Programme level
Full Degree Master

1 block

Block 4
No limitation – unless you register in the late-registration period (BSc and MSc) or as a credit or single subject student.
Study Board of Mathematics and Computer Science
Contracting department
  • Department of Mathematical Sciences
Contracting faculty
  • Faculty of Science
Course Coordinator
  • Niels Martin Møller   (7-58577976766f7c4a776b7e7238757f386e75)
Saved on the 14-02-2024

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Courseinformation of students