Advanced differential geometry

Winter Semester 2018/19

Lecture and exercise class

The lecture takes place wednesday from 2:15–3:45 pm in H2 and roughly every second thursday (on the following dates: Oct 18, Nov 1,8,22, Dec 6, 20, Jan 17, 31) from 10:15–11:45 am in H5. The exercise class takes place roughly every second thursday (on the following dates: Oct 25, Nov 15, 29 Dec 13, Jan 10, 24) from 10:15–11:45 am in H5.

Note that we shifted the course from monday 8:15–9:45 am to wednesday 2:15–3:45 pm!

This lecture is a continuation of the course differential geometry from summer semester 2018 (German-language course description). According to this, basic knowledge in differential geometry and Riemannian geometry (Riemannian manifolds, Levi-Civita connection, geodesics, curvature, Jacobi fields) is preassumed for this lecture.

This lecture consists of a geometrix and an analytic part. In the geometric part, we will try to deduce from local assumptions (on the curvature) assertions about the global structure (topology) of a Riemannian manifold. Assertions of this kind are known as comparison theorems in Riemannian geometry. In the analytic part of the lecture, we will consider properties of Laplace type operators on Riemannian manifolds. In particular, we will show that the Laplace operator of a compact Riemannian manifold admits a discrete spectrum of eigenvalues and that the corresponding eigenfunctions form a complete orthonormal system of the space of quadratically integrable functions. The latter has numerous applications (also in the above mentioned geometric part), not least in Physics (Schrödinger equation of the hydrogen atom).

The exercise sheets can be downloaded in STiNE.