Klaus Kroencke
Lecture and exercise class Advanced differential geometry, Winter term 2018/19
The lecture takes place wednesday from 2:153:45 p.m. in H2 and roughly every second thursday (on the following dates: Oct 18, Nov 1,8,22, Dec 6, 20, Jan 17, 31) from 10:1511:45 a.m. 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 10:1511:45 a.m. in H5.
Note that we shifted the course from monday 8:159:45 a.m. to wednesday 2:153:45 p.m.!
This lecture is a continuation of the course differential geometry from summer term 2018. According to this, basic knowledge in differential geometry and
Riemannian geometry (Riemannian manifolds, LeviCivita 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.
For the preperation of the course, I use the following books and notes:
B. Ammann  Das Yamabe Problem  lecture notes, summer term 2018 (in german) 
M. P. do Carmo  Riemannian Geometry  Mathematics: Theory and Applications 
S. Gallot, Dominique H., J. Lafontaine  Riemannian Geometry  Springer Universitext 
S. Haller  Differentialgeometrie  lecture notes, winter term 2010 and summer term 2011 (in german) 
J. M. Lee  Riemannian Manifolds: An Introduction to Curvature  Springer Graduate Texts in Mathematics 
P. Petersen  Riemannian Geometry  Springer Graduate Texts in Mathematics 
Here are a few books and notes for a further study of elliptic operators on manifolds:
C. Bär  Geometrische Analysis  lecture notes, winter term 2007/08 (in german) 
C. Bär  Geometric Wave Equations  lecture notes, winter term 2015/16 
I. Chavel  Eigenvalues in Riemannian Geometry  Academic Press 
D. Gilbarg, N.S. Trudinger  Elliptic Partial Differential Equations of Second Order  Springer, Gundlehren der mathmatischen Wissenschaften 
H. B. Lawson, M. L. Michelson  Spin Geometry  Princeton University Press 
A. Wassermann  Analysis of Operators  lecture notes, summer term 2009 
