Entry requirements

120 credits including 90 credits in mathematics. Real Analysis or equivalent.

Learning outcomes

In order to pass 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. Compulsory assignments during the course.

Other directives

The course may not be included in the same higher education qualification as Riemannian Geometry (1MA093).