Topics in geometry 2024: negatively curved spaces

Current affairs 28.11.2024

A preliminary final version of the course material has been posted below.

Contents of the course

We begin with an introduction to geodesic metric spaces, In these metric spaces, any two points $x$ and $y$ can be connected with a path whose length coincides with the distance of $x$ and $y$. For example, Euclidean spaces is are geodesic metric spaces, and so are metric graphs. All edges in the following figure have length $1$ and there are exactly three edges meeting at each vertex, imagine that this construction is repeated also for the vertices at the edge of the picture to produce an infinite metric tree.

The hyperbolic plan is a prototype of a negatively curved space. We will see that, in contrast with the flat geometry of the Euclidean plane, triangles in the hyperbolic plane are thin and slim. (We will define quantitative versions of both concepts.)

We study hyperbolic geometry and compare the results with Euclidean geometry to see some differences between flat and negatively curved spaces.

We will then study geodesic Gromov-hyperbolic spaces, defined by requiring all triangles in the space to be thin or slim. Infinite simplicial trees are examples of Gromov hyperbolic spaces that are not manifolds. Triangles in a tree are very thin: any side is contained in the union of the remaining sides.

We will discuss compactification of Gromov hyperbolic spaces by adding a space at infinity. The space at infinity of a Gromov-hyperbolic space $X$ consists of asymptotic classes of geodesic rays (isometric embeddings of the Euclidean half-line) in $X$. This construction gives a geometric meaning for the unit disk as the boundary of the Poincaré disk model of the hyperbolic plane: The space at infinity of the hyperbolic plane is homeomorphic with the unit sphere in the Euclidean plane.

Can you see that the space at infinity of the infinite tree pictured above is a Cantor set?

Time permitting we will discuss further topics towards the end of the course.

Prerequisites

Metric spaces and topology. The geometric concepts needed in the course are introduced in the course material.

Lecture notes

We will follow parts of the text Geometry 2024 (updated 28.11.2024).

Exercises

The exercises are included in the lecture notes. The assigned exercises for each week are listed below.

1	1.2, 1.3, 1.5-1.7
2	6.1-6.5, 6.7
3	7.1-7.4, 7.6, 7.7
4	8.1, 8.3, 8.4, 9.1, 9.2
5	9.5-9.10
6	10.3-10.6

Return your solutions to the problems by the beginning of the exercise class (Monday at 14) as a single pdf file.

Ratkomo

I will be in Ratkomo on Thursdays 14-16.

Contact information

Jouni Parkkonen
Matematiikan ja tilastotieteen laitos
PL 35
40014 Jyväskylän yliopisto