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 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.

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