# Vorlesung "Riemannian Geometry"

Verantwortliche(r): Mattias Dahl

The subject of Riemannian geometry is the study of curvature of spaces. Specifically, it takes the ideas of curvature from curves and surfaces in three-dimensional Euclidean space and generalizes to intrinsic measures of curvature of Riemannian manifolds. These are smooth manifolds equipped with an inner product on the tangent space at each point that varies smoothly with the point.
After a reminder on smooth manifolds, the basic concepts introduced in the course are: the covariant derivative, geodesics, and the Riemannan curvature tensor. The curvature tensor is the fundamental invariant measuring of how curved a Riemannian manifold is. The Jacobi equation tells us how the curvature tensor influences the behaviour of geodesics.
When we have covered the basics of Riemannian geometry we will continue with questions such as: how does curvature influence the global topology of a space? and: how does curvature influence analysis on the space, in particular the study of classical PDEs such as Laplaces equation and the heat equation?
We will try to give many examples illustrating the theory with explicit computations.

The lectures in this course will be held in English. Questions can be asked, and exercise sheets can be handed in, either in German or English. The Exercise classes will be held in German or English.

Wann und Wo:
Mittwoch 14:15-15:45 in 2.09.1.10           (Zeit geändert)
Donnerstag 10:15-11:45 in 2.09.0.14

Übungsgruppe:
Mittwoch 08:30-10:00 in 2.09.0.13 (Florian Hanisch) (Zeit nochmal geändert)

Übungsbetrieb:

Semester (empfohlen):
ab 6.

Geeignet für:
BSc/MSc Mathematik

Modulnummer(n):
771, 772, 781, 81j, MATVMD611, MATVMD612, MATVMD811, MATVMD812, MATVMD813, MATVMD814, MATVMD911, MATVMD912, MATVMD913

Erforderliche Vorkenntnisse:
Analysis 1+2, Elementare Differentialgeometrie

Literatur:
1. C. Bär: Differential geometry. lecture notes, Potsdam 2013
2. C. Bär: Differentialgeometrie. Vorlesungsskript, Potsdam 2006
3. S. Gallot, D. Hulin, J. Lafontaine: Riemannian Geometry. Third edition, Universitext, Springer-Verlag, Berlin 2004
4. M. P. do Carmo: Riemannian Geometry. Birkhäuser, Boston 1992
5. M. Berger: A panoramic view of Riemannian geometry. Springer-Verlag, Berlin 2003