Mathematics, Faculty of Science

Sigmundur Gudmundsson

Riemannian Geometry

Spring 2015

Lecturers: Sigmundur Gudmundsson, ASTRID LINDGREN

Coordinates: Lecture room 332B, Wednesdays at 13:15-16:00

Keywords: differentable manifolds, submanifolds, tangent spaces, tangent bundles, immersions, embeddings, submersions, Riemannian manifolds, the Levi-Civita connection, parallelism, geodesics, the curvature tensor, Jacobi fields and comparison results.

Course Programme: can be found here.

On the 10th of June 1854 Riemann gave his famous "Habilitationsvortrag" in the Colloquium of the Philosophical Faculty at Göttingen. His talk with the title "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" is often said to be the most important in the history of differential geometry. Riemann's revolutionary ideas generalized the geometry of surfaces which had been studied earlier by Gauss, Bolyai and Lobachevsky. Later this lead to an exact definition of the concept of an abstract n-dimensional Riemannian manifold. Gauss, at the age of 76, was in the audience and is said to have been very impressed by his former student.

This course is an introduction to the beautiful theory of Riemannian Geometry, a subject with no lack of interesting examples. They are indeed the key to a good understanding of it and will therefore play a major role throughout the course. Of special interest are the classical Lie groups allowing concrete calculations of many of the abstract notions on the menu.

Literature: No particular textbook will be used but the participants are recommended to have a look at some of the following:

For the history of differential geometry see:

The MacTutor History of Mathematics Archive

Maple routines: curvature.mws