# Differential Geometry II/Differentialgeometrie II

## Summer term 2024

## Current Announcements

Please register on GRIPS. We will send email announcements over this system.
## Language/Sprache

This course is "English on demand". This means the language will be decided
during the first week of the lecture. If there is at least one
non-German-speaking person in the audience
who is sincerely following the lecture, then the lecture will be given in
English.

## Recommended previous knowledge

- Analysis I, II and IV
- Linear Algebra I and II
- Differential geometry I (or comparable)

## Content of the Lecture

This lecture establishes relations between the topology of a smooth manifold
and its curvature properties.
This is a central question within Riemannian geometry. Probably, lecture notes will be written for this lecture.
The lecture builds on the lecture Differential Geometry I, where we
have started to treat the first such relations. The lecture shall lead to a good understand about the consequences of positive or negative curvature, for various notions of curvature (sectional, Ricci, scalar).
We list some examples of relations in the focus of the lecture, including some
relations partially treated at the end of Differential Geometry I.
We will probably not have the time to cover all of them.

- The theorem of Cartan Hadamard: the universal covering of a complete manifold with non-positive sectional curvature is diffeomorphic to ℝ
^{n}
- Bonnet-Myers: the fundamental group of a closed manifold with positive ricci curvature is finite
- Cheeger splitting theorem: a complete manifold with non-negative Ricci-curvature, containing a line, splits as a Riemannian product with the line
- Existence of closed geodesics
- Synge's obstruction to positive sectional curvature
- Structure theorems for ricci-flat manifolds
- Exponential growth of the fundamental group for closed manifolds with negative sectional curvature (Theorem by Milnor)
- Polynomial growth of the fundamental group for closed manifolds with non-negative Ricci curvature (another Theorem by Milnor)
- Special holonomy

Other possible topics of the lecture are Lie groups and vector bundles including characteristic classes.
The precise content of the lecture will depend on the previous knowledge and the interests of the audience.
## Literature

- M. do Carmo, Riemannian Geometry, Birkhäuser
- Cheeger, Ebin, Comparison theorems in Riemannian Geometry
- F. Warner, Foundations of differentiable manifolds and Lie groups, Springer
- T. Sakai, Riemannian Geometry, Transl. Math. Monogr., AMS
- W. Kühnel, Differentialgeometrie, Vieweg
- J. Lee, Introduction to topological manifolds, Springer
- J. Lee, Introduction to smooth manifolds, Springer
- J. Lee, Riemannian manifolds, Springer

## Exercise group

There will be a weekly exercise sheet and a weekly exercise group.
The exercise group is currently planned for Thursday 10-12 (the time and day of week might change!)..

## Exercise Sheets

and finally all exercise sheets in one pdf file.
## Links to related web pages

## Examination/Prüfung

The examination is an oral examination of about 30 minutes.

## Formal criteria/Kriterien für Leistungsnachweise

We refer to the KVV.

Bernd Ammann, 26.01.2024

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