CAT(k) Space

CAT(k) Space

In mathematics, a CAT(k) space, where k is a real number, is a specific type of metric space. Intuitively, triangles in a CAT(k) space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k. In a CAT(k) space, the curvature is bounded from above by k. A notable special case is k = 0: complete CAT(0) spaces are known as Hadamard spaces after the French mathematician Jacques Hadamard.

Originally, Alexandrov called these spaces " domain". The terminology "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).