1. Differentiable manifolds and vector bundles.
2. Linear connections and curvature tensor
3. Riemannian metric, the Levi-Civita connection
5. Geodesics and the exponential map
6. Extremal properties of geodesics
MSc Programme in Mathematics
MSc Programme in Mathematics with a minor subject
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
Academic qualifications equivalent to a BSc degree is recommended.
- 7,5 ECTS
- Type of 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)
- Theory exercises
- Course number
- 7,5 ECTS
- Programme level
- Full Degree Master
- Block 4
- No limit
The number of seats may be reduced in the late registration period
- Study Board of Mathematics and Computer Science
- Department of Mathematical Sciences
- Faculty of Science
- Niels Martin Møller (7-53527471716a77457266796d33707a336970)
- Eric Ling (2-6a71457266796d33707a336970)
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Courseinformation of students