# Topology and curvature of metric spaces

## TL;DR

This paper introduces a novel notion of non-positive curvature for metric spaces based on ball intersection patterns, extending applicability to discrete spaces and linking to topological data analysis.

## Contribution

It proposes a new curvature concept applicable to all metric spaces, including discrete ones, and connects geometric and topological methods via tripod spaces.

## Key findings

- Tripod spaces serve as natural comparison spaces for the new curvature concept.
- Hyperconvex spaces within tripod spaces have trivial Cech homology.
- The approach links metric geometry with persistent homology in data analysis.

## Abstract

We develop a new concept of non-positive curvature for metric spaces, based on intersection patterns of closed balls. In contrast to the synthetic approaches of Alexandrov and Buesemann, our concept also applies to metric spaces that might be discrete. The natural comparison spaces that emerge from our discussion are no longer Euclidean spaces, but rather tripod spaces. These tripod spaces include the hyperconvex spaces which have trivial Cech homology. This suggests a link of our geometrical method to the topological method of persistent homology in topological data analysis. We also investigate the geometry of general tripod spaces.