CAT(0)空间中五个点的一个本质刻画

Pub Date : 2020-01-01 DOI:10.1515/agms-2020-0111
T. Toyoda
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引用次数: 8

摘要

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