# Differential Geometry II/Differentialgeometrie II

## 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 main focus of the lecture is to understand

• Lie groups: these are smooth manifolds with a group structure, such as the orthogonal group O(n), the unitary group U(n) and many more. They are important to decribe symmetries and to construct new examples of Riemannian manifolds
• Some examples of Lorentzian manifolds that are important in general relativity: flat spacetime (=Minkowski space); Schwarzschild spacetime to describe black holes; cosmological aspects
• A deeper understanding of the consequences of positive or negative curvature, for various notions of curvature (sectional, Ricci, scalar). A list of examples in the focus of the lecture, including some relations partially treated at the end of Differential Geometry I are:
• The theorem of Cartan Hadamard: the universal covering of a complete manifold with non-positive sectional curvature is diffeomorphic to ℝn
• 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
The precise content of the lecture will depend on the previous knowledge and the interests of the audience.

## Literature

Lecture Notes will be probably published here.
• 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 in M101.

The first exercise group will take place on Thursday, April 18th, and several notions of basic algebraic toplogy such as, fundamental groups, coverings (Überlagerungen), universal coverings, normal coverings, connections between subgroups of the fundamental groups and coverings, deck transformations, etc. will be recalled and discussed.

## Exercise Sheets

and finally all exercise sheets in one pdf file.

## Examination/Prüfung

The examination is an oral examination of about 30 minutes.

## Formal criteria/Kriterien für Leistungsnachweise

We refer to the KVV.

Bernd Ammann, 23.05.2024
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