CAT(k) space

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

Originally, Alexandrov called these spaces “ domain”. The terminology 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).


Model triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.

For a real number , let denote the unique simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature . Denote by the diameter of , which is if and for .

Let be a geodesic metric space, i.e. a metric space for which every two points can be joined by a geodesic segment, an arc length parametrized continuous curve , whose length

is precisely . Let be a triangle in with geodesic segments as its sides. is said to satisfy the inequality if there is a comparison triangle in the model space , with sides of the same length as the sides of , such that distances between points on are less than or equal to the distances between corresponding points on .

The geodesic metric space is said to be a space if every geodesic triangle in with perimeter less than satisfies the inequality. A (not-necessarily-geodesic) metric space is said to be a space with curvature if every point of has a geodesically convex neighbourhood. A space with curvature may be said to have non-positive curvature.


equipped with the induced length metric is not a space for any .

Hadamard spaces


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 σ1, σ2 are two geodesics in X defined on the same interval of time I, then the function I → R given by

is convex in t.

Properties of spaces

Let be a space. Then the following properties hold:

then .

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See also


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