TY - CHAP

T1 - Gluing theorem and billiards

AU - Alexander, Stephanie

AU - Kapovitch, Vitali

AU - Petrunin, Anton

N1 - Publisher Copyright:
© 2019, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-030-05312-3_2

DO - 10.1007/978-3-030-05312-3_2

M3 - Chapter

AN - SCOPUS:85101116371

T3 - SpringerBriefs in Mathematics

SP - 17

EP - 31

BT - SpringerBriefs in Mathematics

PB - Springer Science and Business Media B.V.

ER -