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
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 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


Course number
7,5 ECTS
Programme level
Full Degree Master

1 block

Block 4
No limit
The number of seats may be reduced in the late registration period
Study Board of Mathematics and Computer Science
Contracting department
  • Department of Mathematical Sciences
Contracting faculty
  • Faculty of Science
Course Coordinator
  • Niels Martin Møller   (7-55547673736c794774687b6f35727c356b72)
Saved on the 28-02-2022

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