Entry requirements

Analysis on Manifolds, Ordinary Differential Equations I

Learning outcomes

In order to pass the course the student should

• be able to define the various geometrical concepts that are introduced in the course, and to use and interpret them in specific examples;
• be familiar with and know how to use central theorems in Riemannian geometry, and be able to give an account of their proofs;
• be able to 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. Morse theory and closed geodesics: critical points of functions and topology of manifolds, spaces of curves in Riemannian manifolds, the theorem of Lyusternik–Fet.

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.