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

Definitions

For a real number k, let M_k denote the unique complete 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 is if 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, 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 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 space if every geodesic triangle \Delta in X with perimeter less than 2D_k satisfies the 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 neighbourhood. A space with curvature \leq 0 may be said to have non-positive curvature.

Examples

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 are two geodesics in X defined on the same interval of time I, then the function I\to \R given by is convex in t.

Properties of CAT(k) spaces

Let (X,d) be a space. Then the following properties hold:

Surfaces of non-positive curvature

In a region where the curvature of the surface satisfies K ≤ 0 , geodesic triangles satisfy the CAT(0) inequalities of comparison geometry, studied by Cartan, Alexandrov and Toponogov, and considered later from a different point of view by Bruhat and Tits. Thanks to the vision of Gromov, this characterisation of non-positive curvature in terms of the underlying metric space has had a profound impact on modern geometry and in particular geometric group theory. Many results known for smooth surfaces and their geodesics, such as Birkhoff's method of constructing geodesics by his curve-shortening process or van Mangoldt and Hadamard's theorem that a simply connected surface of non-positive curvature is homeomorphic to the plane, are equally valid in this more general setting.

Alexandrov's comparison inequality

The simplest form of the comparison inequality, first proved for surfaces by Alexandrov around 1940, states that The inequality follows from the fact that if c(t) describes a geodesic parametrized by arclength and a is a fixed point, then is a convex function, i.e. Taking geodesic polar coordinates with origin at a so that ‖c(t)‖ = r(t) , convexity is equivalent to Changing to normal coordinates u , v at c(t) , this inequality becomes u2 + H−1Hrv2 ≥ 1 , where (u,v) corresponds to the unit vector ċ(t) . This follows from the inequality Hr ≥ H , a consequence of the non-negativity of the derivative of the Wronskian of H and r from Sturm–Liouville theory.