This paper is mainly devoted to proving the four equivalent defining properties of a CAT(κ) space. The proof is based on an interesting tool we established which describes the cyclical five-step deformation procedure for quadrangles in the modal 2-plane S κ 2 , including the limit shape of each step. As a byproduct we give a complete list of cyclical deformation procedures for all kinds of quadrangles on S 1 2 . At last we make a contrast of geometric properties of CAT with CBB spaces, including a comparison between their defining properties and a discussion about Alexandrov’s Lemma.

中文翻译：

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S2κ 四边形的循环变形和 Alexandrov 空间的定义

本文主要致力于证明CAT(κ)空间的四个等价定义性质。该证明基于我们建立的一个有趣的工具，该工具描述了模态 2 平面 S κ 2 中四边形的循环五步变形过程，包括每一步的极限形状。作为副产品，我们给出了 S 1 2 上各种四边形的循环变形过程的完整列表。最后我们将CAT 的几何性质与CBB 空间进行了对比，包括比较它们的定义性质和关于Alexandrov 引理的讨论。