a CAT(0) space
/ kat-ZEER-oh /
Of all the curvature bounds, the most useful single one is 'no more curved than flat space.' A CAT(0) space is a geodesic metric space where every triangle is at least as thin as the corresponding flat Euclidean triangle. Intuitively it is a world with no positive curvature anywhere — like the hyperbolic plane, a tree, a flat sheet, or any combination glued together carefully — and this single condition forces an astonishing amount of good behavior: unique geodesics, convex distances, and a contractible shape.
It is the special case k = 0 of CAT(k), so the model is the flat plane. For any geodesic triangle with vertices p, q, r in X and any points x, y on its sides, d(x, y) <= |x' - y'| where x', y' are the corresponding points of the Euclidean comparison triangle. Three facts follow and are worth knowing as the working definition in practice: (1) between any two points there is a unique geodesic, and it depends continuously on its endpoints; (2) the distance function is convex — if you take two geodesics and connect corresponding-time points, the distance between them is a convex function of time; (3) equivalently, the CN inequality (Bruhat-Tits): for a midpoint m of a geodesic from q to r, d(p, m)^2 <= d(p,q)^2/2 + d(p,r)^2/2 - d(q,r)^2/4. A complete CAT(0) space is called a Hadamard space; by the metric Cartan-Hadamard theorem any complete, simply connected, locally CAT(0) space is globally CAT(0).
CAT(0) spaces are the metric generalization of nonpositively curved manifolds and of the hyperbolic plane, and they are central to geometric group theory: a group acting nicely on a CAT(0) space (a CAT(0) group) inherits a solvable word problem and strong finiteness properties. The convexity of the metric gives a fixed-point theorem — any group of isometries with a bounded orbit fixes the unique circumcenter of that orbit (the Bruhat-Tits fixed point theorem) — which powers rigidity arguments. The most important caveat: CAT(0) is strictly stronger than Gromov-hyperbolic. CAT(0) means thin triangles compared to flat; hyperbolic means thin triangles in a coarser, large-scale sense and allows flat pieces only up to bounded size. The Euclidean plane R^n is CAT(0) but not Gromov-hyperbolic — flatness is allowed under CAT(0) but ruled out at large scale under hyperbolicity.
The hyperbolic plane H^2 (curvature -1 everywhere) is a complete CAT(0) space — in fact CAT(-1), hence also CAT(0). Any two points are joined by a unique geodesic, and triangles are thin: as you move two geodesics rays apart their separation grows, and the distance between corresponding points along two geodesics is a convex function of the parameter. The same convexity makes the nearest-point projection onto any geodesic distance-nonincreasing, a property used constantly in CAT(0) arguments.
The hyperbolic plane is CAT(0): unique geodesics and convex distance between geodesics.
CAT(0) is not the same as Gromov-hyperbolic. R^n is CAT(0) but not hyperbolic; CAT(0) controls local-to-flat curvature, while hyperbolicity is a coarse large-scale condition forbidding big flats. A space can be one without the other.