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.
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 -