An Intrinsic Characterization of Five Points in a CAT(0) Space

Tetsu Toyoda

doi:10.1515/agms-2020-0111

Abstract

Abstract Gromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

Highlights

Under the assumption that a metric space X is geodesic, many simple conditions for X that are equivalent to the condition that X is a CAT( ) space have been known
We prove that a metric space X containing at most ve points admits an isometric embedding into a CAT( ) space if and only if any four points in X satisfy the -inequalities
We introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT( ) space by modifying and generalizing Gromov’s cycle conditions

Summary

Introduction

Under the assumption that a metric space X is geodesic, many simple conditions for X that are equivalent to the condition that X is a CAT( ) space have been known. The following example shows that there exists even a four-point metric space that satis es the quadrilateral inequality (1.1) but does not admit an isometric embedding into any CAT( ) space. Gromov stated in [10, §7] that a four-point metric space admits an isometric embedding into a CAT( ) space if and only if it satis es the -inequalities (see Theorem 1.7 below).

Results

Conclusion