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

Education

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

 

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.

Lectures and tutorials:

3+2 lectures (including possible 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.

Written
Oral
Continuous feedback during the course of the semester
ECTS
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.
Aid
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