NMAK20006U Riemannian Geometry

Volume 2022/2023
Education

MSc Programme in Mathematics
MSc Programme in Mathematics with a minor subject

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

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

 

Skills:

  • 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.

Lecture notes and/or textbook

Geometri 2 or corresponding knowledge of differentiable manifolds

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

3+2 lectures (including seminars by students) and 2+2 tutorials per week during 8 weeks.
  • Category
  • Hours
  • Lectures
  • 40
  • Preparation
  • 106
  • Theory exercises
  • 32
  • Exam
  • 28
  • Total
  • 206
Written
Oral
Continuous feedback during the course of the semester
Credit
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.
Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner
Re-exam

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 assesment

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