Gluing theorem and billiards

Stephanie Alexander, Vitali Kapovitch, Anton Petrunin

Research output: Chapter in Book/Report/Conference proceedingChapter


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.

Original languageEnglish (US)
Title of host publicationSpringerBriefs in Mathematics
PublisherSpringer Science and Business Media B.V.
Number of pages15
StatePublished - 2019

Publication series

NameSpringerBriefs in Mathematics
ISSN (Print)2191-8198
ISSN (Electronic)2191-8201

All Science Journal Classification (ASJC) codes

  • General Mathematics


Dive into the research topics of 'Gluing theorem and billiards'. Together they form a unique fingerprint.

Cite this