: Mathématiques: Cartan-Alexandrov-Toponogov space

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geometry > topological space > metric space > Cartan-Alexandrov-Toponogov space

Cartan-Alexandrov-Toponogov space

In mathematics, a ${\\displaystyle \\mathbf {\\operatorname {\ extbf {CAT}} } (k)}$ space, where ${\\displaystyle k}$ is a real number, is a specific type of metric space. Intuitively, triangles in a ${\\displaystyle \\operatorname {CAT} (k)}$ space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature ${\\displaystyle k}$ . In a ${\\displaystyle \\operatorname {CAT} (k)}$ space, the curvature is bounded from above by ${\\displaystyle k}$ . A notable special case is ${\\displaystyle k=0}$ ; complete ${\\displaystyle \\operatorname {CAT} (0)}$ spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/CAT(k)_space)

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