Geometry 2
3-4 hours of lectures per week.
Lecturer
Marcel Bökstedt.
Content
In Topology 1,
n-dimensional manifolds are defined. These are generalizations of the 2-dimensional surfaces studied in Geometry 1. We study the generalizations of geometrical concepts introduced in Geometry 1 to
n-dimensional manifolds. Examples include geodesics and various forms of curvature, including the Ricci curvature used in Perelman's recently announced proof of the Poincarè conjecture.
The most important results will relate local structure of a Riemannian manifold to global structure. We will prove that under obvious local conditions there always exists a shortest path between two points, and that it is a geodesic. We will also prove that a simply connected manifold of negative sectional curvature is diffeomorphic to $\R^n$.
Prerequisites
Geometry 1 and Topology 1.
Literature
John M. Lee,
Riemannian Manifolds, An Introduction to Curvature, Springer Graduate Texts in Mathematics 176, New York, 1997.
Evaluation
Students who do not intend to take a degree in Mathematics or Statistics from the University of Aarhus, but wish to earn credits for a 2.dels course from the Department of Mathematics, should indicate at the beginning of the course that they wish to be examined.
The form of examination for these students will be active participation together with oral or written contributions.
ECTS-point
10
Quarter
Spring 2004