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
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 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.
- ECTS
- 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
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
Kursusinformation
- Language
- English
- Course number
- NMAK20006U
- ECTS
- 7,5 ECTS
- Programme level
- Full Degree Master
- Duration
-
1 block
- Placement
- Block 4
- Schedulegroup
-
A
- Capacity
- No limit
The number of seats may be reduced in the late registration period - Studyboard
- Study Board of Mathematics and Computer Science
Contracting department
- Department of Mathematical Sciences
Contracting faculty
- Faculty of Science
Course Coordinator
- Niels Martin Møller (7-62618380807986548175887c427f8942787f)
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Courseinformation of students