3-4 hours of lectures per week.
- Lecturer:
- Marcel Bøkstedt and Jørgen Tornehave
- Content:
- This course is an introduction to Riemannian manifolds. These are higher dimensional versions of the surfaces studied in Geometry 1. Part of the material can be considered as continuations of themes studied in that course. We introduce fundamental concepts like the curvature tensor, which generalizes the Gauss curvature of surfaces. We also discuss geodesic curves, the exponential map and sectional curvature. We will give various examples, for instance the hyperbolic spaces, which are the spaces of higher dimensional non-Euclidean geometry. Theorems we will prove include that a geodesic curve is locally a shortest path, and we examine under what conditions a geodesic curve can be continued indefinitely. We will prove that a simply connected manifold with negative curvature (under reasonable technical conditions) is diffeomorphic to Euclidean space.
- Prerequisites:
- Topology 1.
- Literature:
- Peter Petersen: "Riemannian geometry" Graduate Texts in Mathematics 171. Springer 1997.