CAT(0)空间的温和介绍
A Gentle Introduction to CAT(0) Spaces
AI总结：
本文通过探讨度量空间的几何特性，介绍如何定义曲线长度和测地线，并通过与模型空间的比较来定义非正曲率。重点介绍了CAT(0)空间的性质及其独特优势，为读者提供一个简洁且自包含的入门指南。
摘要：
在本项目中，我们探索一般度量空间的几何特性，其中我们并不一定拥有微分几何的工具。某些度量空间$(X,d)$允许我们定义测地线，使我们能够将$(X,d)$中的测地三角形与所谓的模型空间中的测地三角形进行比较。第一章和第二章首先讨论如何定义曲线长度和$(X,d)$上的测地线，然后利用这些来阐述度量空间的“非正曲率”概念。第三章关注这类非正曲率度量空间的特殊情况，称为CAT(0)空间。这些空间具有特别良好的性质，例如唯一测地性、可缩性以及凸度量等。我们主要参考了Martin R. Bridson和André Haefliger的书籍，但有一些不同。首先，我们仅使用欧几里得平面$\mathbb{E}^2$作为模型空间，这足以定义CAT(0)空间。其次，我们跳过了许多章节，因为它们对我们的特定目的不相关。最后，我们为一些证明添加了细节，这些细节在原始文献中可能稀少或完全不存在。通过这种方式，我们希望为CAT(0)空间创建一个更加流畅、自包含且易于理解的入门指南。
英文摘要：
In this project we explore the geometry of general metric spaces, where we do not necessarily have the tools of differential geometry on our side. Some metric spaces $(X,d)$ allow us to define geodesics, permitting us to compare geodesic triangles in $(X,d)$ to geodesic triangles in a so called model space. In Chapters 1 and 2 we first discuss how to define the length of curves, and geodesics on $(X,d)$, and then using these to portray the notion of "non-positive curvature" for a metric space. Chapter 3 concerns itself with special cases of such non-positively curved metric spaces, called CAT(0) spaces. These satisfy particularly nice properties, such as being uniquely geodesic, contractible, and having a convex metric, among others. We mainly follow the book by Martin R. Bridson and André Haefliger, with some differences. Firstly, we restrict ourselves to using the Euclidean plane $\mathbb{E}^2$ as our model space, which is all that is necessary to define CAT(0) spaces. Secondly, we skip many sections of the mentioned book, as many are not relevant for our specific purposes. Finally, we add details to some of the proofs, which can be sparse in details or completely non-existent in the original literature. In this way we hope to create a more streamlined, self-contained, and accessible introduction to CAT(0) spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
