$\mathrm{CAT}(0)$空间中五点的一个内在刻画
An intrinsic characterization of five points in a $\mathrm{CAT}(0)$ space
摘要：
Gromov（2001）和Sturm（2003）证明了$\mathrm{CAT}(0)$空间中的任意四个点满足一定的不等式族。我们称这些不等式为$\boxtimes$ -不等式，遵循Gromov使用的符号。本文证明了最多包含五个点的度量空间$X$允许等距嵌入到$\mathrm{CAT}(0)$空间中当且仅当$X$中的任何四个点满足$\boxtimes$ -不等式。为了证明这一点，我们引入了一个新的家庭的必要条件的度量空间允许等距嵌入到$\mathrm{CAT}(0)$空间通过修改和推广格罗莫夫的循环条件。进一步证明了如果度量空间满足所有这些必要条件，则它允许等距嵌入到$\mathrm{CAT}(0)$空间中。这项工作提出了一种新的方法来表征这些度量空间，允许等距嵌入到$\mathrm{CAT}(0)$空间。
英文摘要：
Gromov (2001) and Sturm (2003) proved that any four points in a $\mathrm{CAT}(0)$ space satisfy a certain family of inequalities. We call those inequalities the $\boxtimes$-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 $\mathrm{CAT}(0)$ space if and only if any four points in $X$ satisfy the $\boxtimes$-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a $\mathrm{CAT}(0)$ 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 $\mathrm{CAT}(0)$ space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a $\mathrm{CAT}(0)$ space.
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