# SF3674 Differential geometry, graduate course, fall 2016

## Description

This is an overview course targeted at all graduate students in mathematics. The goal is to give an introduction to some of the methods and research areas of modern differential geometry.

Prerequisities are preferably some introductory course on differential manifolds, and advanced level courses on algebra, analysis, and topology

## Lecturers

## Time and place

The lectures takes place* Tuesdays 15:15-17:00* in room 3418 at the math department.

## Content

Introduction

- Manifolds, semi-Riemannian metrics, curvature, submanifolds, hyperquadrics, geodesics, comparison theorems for positive/negative curvature.

Followed by a selection from the subjects

- Curvature in general, holonomy, characteristic classes
- Lorentzian geometry, Hawking-Penrose singularity theorems
- Lie groups, homogeneous spaces, symmetric spaces
- Morse theory, De Rham cohomology
- Elliptic operators, spectral geometry, index theory
- Vector fields, distributions, foliations, differential systems, Frobenius theorem
- General relativity
- ...

## Course literature

- Barrett O'Neill, "Semi-Riemannian Geometry"
- Christian Bär, lecture notes
- ...

## Lectures

Here reading instructions and exercises for each lecture.

- Tuesday September 6
- Tuesday September 13
- Tuesday September 20
- Tuesday September 27
- Tuesday October 4
- Tuesday October 11
- Tuesday October 18

## Examination

Homework problems and oral test/presentation.