Geometry 2
3-4 hours of lectures per week.
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.
Fundamental concepts are introduced, like the curvature of surfaces.
We also discuss geodesic curves, the exponential map and sectional
curvature.
These are various examples, for instance the hyperbolic spaces, which are
the spaces of higher dimensional non-Euclidean geometry.
Theorems will include that a geodesic curve is locally a shortest path,
and we will examine under what conditions a geodesic curve can be continued
indefinitely.
Another subject which might be studied in the course is the geometry of symmetric
spaces.
Prerequisites Geometry 1 and Topology 1. The intention is to make it possible to follow the
courses Topology 1 and Geometry 2 simultaneously.
Text-books Johan Dupont: Differential Geometry (Lecture notes No 62, Århus)
ECTS-point 10