CAT(k) space

In mathematics, a ${\displaystyle \mathbf {\operatorname {\textbf {CAT}} } (k)}$ space, where ${\displaystyle k}$ is a real number, is a specific type of metric space. Intuitively, triangles in a ${\displaystyle \operatorname {CAT} (k)}$ space (with ${\displaystyle k<0}$) 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.

Originally, Aleksandrov called these spaces “${\displaystyle {\mathfrak {R}}_{k}}$ domains”. The terminology ${\displaystyle \operatorname {CAT} (k)}$ was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications).