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Sommersemester 2013

Seminar: Topics in Geometry and Analysis

Gerasim Kokarev (Mathematisches Institut der LMU)

with the assistance of
Asma Hassannezhad (Mathematisches Institut der LMU)

This is a working seminar on differential geometry and analysis. There is a main topic for the semester with possibly additional talks on other topics.

The main topic for the current semester is Sub-Riemannian geometry. This seminar is an ideal supplement to the courses on Riemannian geometry and Metric geometry, and is intended for MSc students in Mathematics and Physics.

We shall treat the basics of sub-Riemannian geometry, focusing on the properties of Carnot-Caratheodory metric, the notion of tangent space, and Carnot groups.

The core module "Differenzierbare Mannigfaltigkeiten/Differential geometry".

Time and venue:
14.00-16.00 Thursday, Room B046

Tentative programme:

18 April 2013
Organisational meeting

25 April 2013 (G. Kokarev)
Sub-Riemannian manifolds and Carnot-Caratheodory metrics; theorems of Chow and Sussmann.
[1], sections 1.1-2.1.

2 May 2013 (A. Hassannezhad)
Topological properties of Carnot-Caratheodory distance, Hopf-Rinow theorem, effective proof of Chow's theorem. [1], sections 2.2-2.5

9 May 2013

16 May 2013 (G. Kokarev)
Grushin plane and Heisenberg group. [1], section 3; [3], sections 1.5, 2.5.

23 May 2013 (L. Schiemanowski)
Privileged coordinates. [1], section 4.

30 May 2013

6 June 2013
kein Vortrag

13 June 2013 (A. Hassannezhad)
Nilpotent Lie groups and nilpotent Lie algebras; Baker-Campbell-Hausdorff formula; Carnot groups. [3], sections 3.1-3.2; [5] sections 1.1-1.2.

20 June 2013 (A. Hassannezhad)
The tangent nilpotent Lie algebra and the algebraic structure of the tangent space.
[1], section 5; [3], section 3.3.

27 June 2013 (L. Schiemanowski)
Gromov's notion of the tangent space; distance estimates and the metric tangent space.
[1], sections 6-7.

4 July 2013 (A. Ananin)
Basic notions of CR geometry: Levi form, Levi distribution, model spaces (Heisenberg group, CR sphere, boundary of the Siegel domain), volume form, sub-Laplacian. [2], sections 1.1, 2.1.

11 July 2013 (A. Doicu)
Tanaka-Webster connection and curvature theory on CR manifolds; CR space forms.
[2], sections 1.2, 1.4.

Reading list:
1. Bellaiche, A. The tangent space in sub-Riemannian geometry. Sub-Riemannian geometry, 1-78, Progr. Math., 144, Birkhäuser, Basel, 1996.
2. Dragomir, S., Tomassini, G. Differential geometry and analysis on CR manifolds. Progr. Math., 246. Birkhäuser Boston, Inc., Boston, MA, 2006. xvi+487 pp.
3. Le Donne. E., Lecture notes on sub-Riemannian geometry. Preprint, 2010.
4. Montgomery, R., A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs, 91. AMS, Providence, RI, 2002.
5. Corwin, L. J., Greenleaf, F. P. Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples. Cambridge Studies in Advanced Mathematics, 18. Cambridge University Press, Cambridge, 1990.

Last modified: 22 March 2013