Entry requirements

Real Analysis, Ordinary Differential Equations I

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 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 and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.