a CAT(k) space
/ kat-kappa /
Take the opposite intuition to a convex sphere: imagine a saddle, or the inside of a tree-like branching network, where triangles are thin and skinny, spreading apart faster than in flat space. A CAT(k) space captures spaces that are at most as curved as the model of constant curvature k — 'curvature bounded above.' The acronym honors Cartan, Alexandrov, and Toponogov, the three names attached to the comparison idea. It is the single clean definition of upper curvature bounds that works on any geodesic metric space, smooth or not.
Fix a model curvature k and the model surface M_k. A geodesic metric space X is CAT(k) if every geodesic triangle in X (with perimeter below 2pi/sqrt(k) when k > 0) is at least as thin as its comparison triangle in M_k. Precisely: for a triangle with vertices p, q, r, and for any two points x, y on its sides, d(x, y) <= |x' - y'|, where x', y' are the corresponding comparison points in M_k. So the real triangle never bulges out past the flat (or spherical, or hyperbolic) model — distances across a triangle are no greater than in the model. Equivalently, every vertex angle is no larger than its comparison angle. One small but vital consequence baked into being CAT(k): in CAT(k) spaces (with the perimeter restriction when k > 0) geodesics between two points are unique, and they vary continuously with their endpoints.
Upper curvature bounds tame a space: CAT(k) spaces are locally contractible, have unique local geodesics, and inherit much of the rigidity of nonpositively or boundedly-curved manifolds. They are the metric backbone for building objects out of pieces — CAT(k) is a local condition that can be checked on a complex glued from flat or spherical cells via Gromov's link condition (the link of each vertex must itself have a curvature bound). A crucial caveat about the model parameter: a CAT(k) space is automatically CAT(k') for every k' >= k (a smaller upper bound is a stronger statement), and CAT(0) is the most important special case, but CAT(1) and CAT(k) for positive k are essential when you want spaces that look locally spherical. Do not confuse CAT(k) (upper bound, thin triangles) with Alexandrov curvature >= k (lower bound, fat triangles).
A metric tree — a connected graph with no loops, each edge a real interval, distance measured along the unique path between points — is CAT(k) for every k, and in particular CAT(0). Any 'triangle' degenerates: the geodesic between two points passes through their nearest common branch point, so the three sides of a triangle overlap into a tripod with no area. Such a degenerate triangle is thinner than every comparison triangle, which is why trees are the extreme case of nonpositive (indeed arbitrarily negative) curvature.
A metric tree is CAT(0): triangles collapse to tripods, thinner than any flat comparison triangle.
CAT(k) is local in a precise sense only after care: a complete CAT(0) space is globally so (Cartan-Hadamard for metric spaces), but for k > 0 'locally CAT(k)' does not upgrade to global without a bound on geodesic length. Always state which k and whether the claim is local or global.