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Sommersemester 2012

Riemannian Geometry

Gerasim Kokarev (Mathematisches Institut der LMU)

Course description:
This course provides an introduction to basic concepts in Riemanian geometry, which have important links with other mathematical and physical disciplines.

It is oriented on students in Mathematics and Physics, and is one of the modules in the Mathematics Master Programme as well as the Master Programme in Theoretical and Mathematical Physics (TMP).

The core module "Differenzierbare Mannigfaltigkeiten/Differential geometry". The knowledge of the modules "Geometrie und Topologie von Flächen" and "Topology I" is beneficial, but not necessary.

Lectures schedule:
Lectures will be given in English twice a week, 10.00-12.00 Tue and 10.00-12.00 Fri, Room B 040

Course outline:
The course includes the standard material on geometry of hypersurfaces, geodesics, distance functions, basic comparison theorems, and relationships between topology and geometry.

Reading list:
1. Chavel, I. Riemannian geometry. A modern introduction. Second edition. Cambridge Studies in Advanced Mathematics, 98. Cambridge University Press, Cambridge, 2006. xvi+471 pp.
2. Cheeger, J., Ebin, D. Comparison theorems in Riemannian geometry. North-Holland Mathematical Library, Vol. 9. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. viii+174 pp.
3. Petersen, P. Riemannian geometry. Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998. xvi+432 pp.

Exercise classes (Übungen):
The exercise classes will hold at 10.00-12.00 Wed, Room B040.

The problem sets for the exercise classes are typed and posted by Robert Schmidt on this page on Fridays and are due a week later by Friday 1pm; please leave sheets with your solutions in the drop-in box on the first floor marked "Riemannian geometry". Students are encouraged to do as many problems in these sets as possible, since they comprise an important part of the course and similar problems are likely to appear on the exam.

Mid-Term Test:
The mid-term test will hold on the 22th of June at 10.00-12.00, Room B040. Please bring your student ID along with a photo ID, and make sure that you arrive before 10.00am.

Students are encouraged to take the test, as it gives a good idea of their preparation level and the kind of exercises that will be given on the final exam. Scores over 60% will be taken into account for the final grade.

Final Exam:
The final exam will hold on the 24th of July at 10.00-12.00, Room B040. Please bring your student ID along with a photo ID, and make sure that you arrive before 10.00am.

Course Programme:

1. Introduction I

1.1. Riemannian manifolds; examples and constructions
1.2. Characteristics defined by Riemannian metrics: volume, length, and distance
1.3. Relationships between distance and volume: Caratheodory and Gromov theorems

2. Geodesics

2.1. Background on connections and parallel transport
2.2. Geodesics and exponential map: basic properties
2.3. Geodesics are locally shortest; Gauss lemma; other applications
2.4. Geodesics and completeness: Hopf-Rinow theorem

3. Introduction II

3.1. Background on Riemann curvature tensor; constant curvature spaces
3.2. Submanifolds in Riemannian manifolds: Gauss and Codazzi-Mainardi equations
3.3. Hypersurfaces in Euclidean spaces: relationship between curvature and shape operators, its consequences. Hopf theorem.
3.4. Pseudo-distance functions and geometry of their level sets: fundamental equations and their applications.

4. Variational theory of geodesics

4.1. First and second variation of the length
4.2. Jacobi fields, conjugate points, and the Jacobi theorem on the index form
4.3. Applications: Cartan-Hadamard, Bonnet-Myers, critical points of exp, Riemann normal coordinates, local isometries between constant curvature spaces

5. Riemannian coverings and comparison theorems

5.1. Local isometries and Riemannian coverings; Killing-Hopf theorem.
5.2. Group actions by isometries; characteristics of isometries (displacement function, period, axis). Application to the existence of closed geodesics.
5.3. Geometry and topology of non-positively curved manifolds: topological Cartan-Hadamard, convex functions, fixed points of isometries (Cartan theorem), triangle comparisons, Preissmann theorem.
5.4. Non-negatively curved manifolds: Synge's and Toponogov theorems.

6. Riemannian volume and comparison theorems

6.1 Pre-requisities on the geodesic spherical coordinates and cut locus. Riemannian volume of metric balls and spheres; lower area function.
6.2 Volume comparison theorems: Günther-Bishop (sectional curvature comparison), Bishop (Ricci curvature comparison), and Gromov's relative volume comparison. Application: Toponogov-Cheng diameter theorem.
6.3 Volume growth and growth of groups: basic facts, examples, and relationships (Svarc-Milnor). Growth of groups and curvature.

Last modified: 28 June 2011