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Published in:
2019 | OriginalPaper | Chapter
2. Gluing theorem and billiards
Authors : Stephanie Alexander, Vitali Kapovitch, Anton Petrunin
Published in: An Invitation to Alexandrov Geometry
Publisher: Springer International Publishing
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Abstract
In this chapter we define \(\mathrm{CAT}^{}(\upkappa )\) spaces and give the first application, to billiards. Here “\(\text {CAT}\)” is an acronym for Cartan, Alexandrov, and Toponogov. It was coined by Mikhael Gromov in 1987. Originally, Alexandrov called these spaces “\(\mathfrak {R}_\upkappa \) domain”; this term is still in use. Riemannian manifolds with nonpositive sectional curvature provide a motivating example. Specifically, a Riemannian manifold has nonpositive sectional curvature if and only if each point admits a \(\mathrm{CAT}^{}(0)\) neighborhood.