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In mathematics, a \mathbf{\operatorname{\textbf{CAT}}(k)} space, where k is a real number, is a specific type of metric space. Intuitively, triangles in a \operatorname{CAT}(k) space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k. In a \operatorname{CAT}(k) space, the curvature is bounded from above by k. A notable special case is k=0 complete \operatorname{CAT}(0) spaces are known as Hadamard spaces after the French mathematician Jacques Hadamard.
Originally, Alexandrov called these spaces “\mathfrak{R}_k domain”. The terminology \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).
For a real number k, let M_k denote the unique simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature k. Denote by D_k the diameter of M_k, which is +\infty if k \leq 0 and \frac{\pi}{\sqrt{k}} for k>0.
Let (X,d) be a geodesic metric space, i.e. a metric space for which every two points x,y\in X can be joined by a geodesic segment, an arc length parametrized continuous curve \gamma\,:\,[a,b] \to X,\ \gamma(a) = x,\ \gamma(b) = y, whose length
is precisely d(x,y). Let \Delta be a triangle in X with geodesic segments as its sides. \Delta is said to satisfy the \mathbf{\operatorname{\textbf{CAT}}(k)} inequality if there is a comparison triangle \Delta' in the model space M_k, with sides of the same length as the sides of \Delta, such that distances between points on \Delta are less than or equal to the distances between corresponding points on \Delta'.
The geodesic metric space (X,d) is said to be a \mathbf{\operatorname{\textbf{CAT}}(k)} space if every geodesic triangle \Delta in X with perimeter less than 2D_k satisfies the \operatorname{CAT}(k) inequality. A (not-necessarily-geodesic) metric space (X,\,d) is said to be a space with curvature \leq k if every point of X has a geodesically convex \operatorname{CAT}(k) neighbourhood. A space with curvature \leq 0 may be said to have non-positive curvature.
As a special case, a complete CAT(0) space is also known as a Hadamard space; this is by analogy with the situation for Hadamard manifolds. A Hadamard space is contractible (it has the homotopy type of a single point) and, between any two points of a Hadamard space, there is a unique geodesic segment connecting them (in fact, both properties also hold for general, possibly incomplete, CAT(0) spaces). Most importantly, distance functions in Hadamard spaces are convex: if σ_{1}, σ_{2} are two geodesics in X defined on the same interval of time I, then the function I → R given by
is convex in t.
Let (X,d) be a \operatorname{CAT}(k) space. Then the following properties hold:
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