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

IF 0.9 3区数学 Q2 MATHEMATICS

Analysis and Geometry in Metric Spaces Pub Date : 2020-01-01 DOI:10.1515/agms-2020-0111

T. Toyoda

引用次数: 8

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.

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CAT（0）空间中五个点的一个本质刻画

Gromov（2001）和Sturm（2003）证明了CAT（0）空间中的任意四个点满足一个不等式族。我们将这些不等式称为⊠-不等式，遵循Gromov使用的符号。在本文中，我们证明了包含最多五个点的度量空间X允许等距嵌入到CAT（0）空间中，当且仅当X中的任意四个点满足⊠-不等式。为了证明这一点，我们通过修改和推广Gromov的循环条件，引入了度量空间允许等距嵌入到CAT（0）空间的一个新的必要条件族。此外，我们证明了如果度量空间满足所有这些必要条件，那么它允许等距嵌入到CAT（0）空间中。这项工作提出了一种新的方法来表征那些允许等距嵌入到CAT（0）空间中的度量空间。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

16 weeks

期刊介绍： Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.