A Gentle Introduction to CAT(0) Spaces

TL;DR

This paper provides a streamlined, accessible introduction to CAT(0) spaces, focusing on their geometric properties and differences from general metric spaces, based on a simplified approach from Bridson and Haefliger's work.

Contribution

It offers a simplified, self-contained explanation of CAT(0) spaces, emphasizing Euclidean plane models and adding detailed proofs for clarity.

Findings

CAT(0) spaces are uniquely geodesic and contractible.

They have convex metrics and special geometric properties.

The approach simplifies understanding of non-positive curvature in metric spaces.

Abstract

In this project we explore the geometry of general metric spaces, where we do not necessarily have the tools of differential geometry on our side. Some metric spaces $(X,d)$ allow us to define geodesics, permitting us to compare geodesic triangles in $(X,d)$ to geodesic triangles in a so called model space. In Chapters 1 and 2 we first discuss how to define the length of curves, and geodesics on $(X,d)$, and then using these to portray the notion of "non-positive curvature" for a metric space. Chapter 3 concerns itself with special cases of such non-positively curved metric spaces, called CAT(0) spaces. These satisfy particularly nice properties, such as being uniquely geodesic, contractible, and having a convex metric, among others. We mainly follow the book by Martin R. Bridson and Andr\'e Haefliger, with some differences. Firstly, we restrict ourselves to using the Euclidean plane $\mathbb{E}^2$ as our model space, which is all that is necessary to define CAT(0) spaces. Secondly, we skip many sections of the mentioned book, as many are not relevant for our specific purposes. Finally, we add details to some of the proofs, which can be sparse in details or completely non-existent in the original literature. In this way we hope to create a more streamlined, self-contained, and accessible introduction to CAT(0) spaces.