Riemannian Geometry
Syllabus, Master's level, 1MA196
- Code
- 1MA196
- Education cycle
- Second cycle
- Main field(s) of study and in-depth level
- Mathematics A1N, Mathematics A1N
- Grading system
- Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
- Finalised by
- The Faculty Board of Science and Technology, 18 February 2021
- Responsible department
- Department of Mathematics
Entry requirements
120 credits including 90 credits in mathematics. Real Analysis or the equivalent. Proficiency in English equivalent to the Swedish upper secondary course English 6.
Learning outcomes
On completion of the course, the student should be able to:
- define the various geometrical and algebraic concepts that are introduced in the course, and be able to use and interpret them i specific examples;
- use and formulate central theorems in Riemannian geometry and Topology, and be able to give an account of their proofs.
- use the theory, methods and techniques of the course to solve problems.
Content
Parallel transport: connections, covariant derivate, curvature. The Yang-Mills functional, Levi-Civita connections. Geodesics: first and second variations of arc length, Jacobi fields, conjugate points, comparison theorems. The fundamental group, the theorem of Seifert-van Kampen, existence theorems of geodesics, spaces of curves in Riemannian manifolds.
Instruction
Lectures and problem solving sessions.
Assessment
Oral examination. Written assignments during the course in accordance with instructions at the beginning of the course.
If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.
Other directives
The course may not be included in the same higher education qualification as Riemannian Geometry (1MA093).