The class is devoted to an introduction to harmonic maps and
their use in differential geometry, in particular
in the case that the domain of the map is a closed surface
of higher genus. The Hopf differential of such a map will be introduced.
Existence results
when the target of the map is nonpositively curved will be covered,
the energy function will be studied as a function on
Teichmüller space of conformal structures on the domain using the Hopf
differential. Various applications to the relation between
geometry and topology will be discussed.
The Ricci flow will be introduced and studied in the case of surfaces.
Prerequisits: Basic knowledge in Riemannian geometry and in topology,
multivariable calculus. Extended knowledge in PDEs is useful but
not necessary.
Topics covered are Lie groups, principal bundles, connections,
the Chern-Weil homomorphism, Chern classes, Pontrjagin class and the
Euler class, applications.
Prerequisits: The material covered in the class "Foundations of
analysis and geometry on manifolds"
Literature:
Johan Dupont, Fiber bundles and Chern-Weil theory, Lecture Notes,
Aarhus University 2003
S. Kobayashi, K. Nomizu, Foundations of differential geometry II,
Wiley
John Lee, Introduction to smooth manifolds, Springer Graduate Texts
The seminar will be held in person
Wednesday, 12h-14h, Kleiner Hörsaal
Begin: April 6
1) Connections on vector bundles
2) The Levi Civita connection of a Riemannian metric
3) Geodesics
4) Completeness
5) Curvature
6) The Laplacian
7) Basic properties of Lie groups
8) Spaces of constant curvature
Prerequisits:
The content of the class "Foundations of analysis and geometry on manifolds"
The class will largely follow the book
John M. Lee, Introduction to Riemannian manifolds, Springer Grad. Text.
The seminar will be held in hybrid form:
Friday, 14h-16, SR 0.011
Grundzüge der Analysis und Geometrie auf Mannigfaltigkeiten V3D3
with Jonas Beyrer
Topics covered:
1) Smooth manifolds and their tangent bundle
2) Smooth maps between manifolds
3) Submersions and immersions
4) Vector fields, integral curves and flows
5) Vector bundles
6) Differential forms and their integration
7) de Rham cohomology
8) Riemannian metrics
Prerequisits:
A solid knowlege on basic analysis (in particular the implicit function
theorem)
Basic point set topology
Fundamental group
The class will largely follow the book
John M. Lee, Introduction to smooth manifolds, Springer Grad. Text 2013.
At the moment the class is planned to be held in person. Time:
Tu 14.15h-16h, Großer Hörsaal
Fr 12.15h-14h, Großer Hörsaal
Begin: Tu, October 12
The lectures can also be attended online. More information is available
on e-campus. Please register on e-campus for the class.
Tutorials:
Group 1: We 8h-10h online, tutor David Aretz
Group 2: We 16h-18h Kleiner Hoersaal We 10, tutor Alvaro Sanchez Hernandez
Group 3: Th 14h-16h Zeichensaal We 10, tutor Nil Rodellas Gracia
Group 4: Fr 10h-12h online, tutor Lory Kadiyan
Courses SS 21
Advanced topics in differential geometry V5D5
This is a continuation of the lecture Advanced Geometry I from
the winter semester. There will however be no tutorials.
Time: We 12.15h via Zoom.
The Zoom information is available on e-campus.
Begin: April 21, 2021.
Courses WS 20/21
Advanced Geometry I V4D3
The topic of the class is an introduction to finiteness results
in Riemannian geometry.
The basic questions that we can ask are:
Can we understand all smooth closed (compact without boundary)
Riemannian manifolds with suitably chosen geometric constraints?
What are the geometric constraints that we need to impose to
make this class both interesting (in particular, not empty)
and controllable
(for example, finite, or compact in a suitable sense),
and what is the impact of the geometric constraints on the topology?
The class will be organized in two parts, the second part is
Advanced Geometry II in the Summer Semester 2021.
The goal of the first part will be to
- study relevant classes of
examples
- understand what the correct questions are
- introduce some foundational concects like convergence and
collapsing.
More concretely, the topics covered are:
1) Surfaces and the Gauss-Bonnet theorem
2) An short introduction to Lie groups, symmetric spaces
and homogeneous spaces
3) Volume and volume comparison, the Bishop-Gromov theorem
4) Fundamental groups and Ricci curvature
5) Gromov-Hausdorff convergence
6) Finiteness results of Cheeger and Gromov
Literature:
John Lee, Introduction to Riemannian manifolds, Springer Grad. Text 2018
Jeff Cheeger, David Ebin, Comparison theorems in Riemannian geometry,
North Holland 1975
Peter Petersen. Riemannian geometry, Springer 1998.
M. Gromov, Metric structures on Riemannian and non-Riemannian spaces,
Birkhäuser
Prerequisits: Basic knowledge in Riemannian geometry
The class will be held online, via Zoom.
Time: Tu, 14.15h, Fr, 12.15h
Begin: 27.10.2020
More information on the class including the access code for Zoom
is available on ecampus.
Informal registration on ecampus for the class is necessary.
The password is available on Basis.
Textbook: John Lee, Introduction to Riemannian manifolds,
second edition, Springer Graduate Texts in Math., Springer 2018
Hauptseminar Geometrie S2D1
Graduate Seminar on Differential Geometry S4D1
with Matteo Costantini
Topic: Foliations
Literature:
C. Camacho, A. Lins Neto, Geometric theory of foliations,
Birkhäuser 1985
Y. Eliashberg, W. Thurton, Confoliations, University Lecture Series,
AMS 1998.