Entry requirements

Real Analysis, Ordinary Differential Equations I. 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 in 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 derivative, curvature. The Yang-Mill functional, Levi-Cevita 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

Written examination at the end of the course combined with compulsory assignments during 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.