CAT(k) space

Definition

Let $k$ be a real number. Let $M_k$ be the complete simply connected surface of constant curvature $k$ . It is the round sphere of radius $1/\sqrt{k}$ if $k > 0$ , the Euclidean plane if $k = 0$ , and the hyperbolic plane of curvature $k$ if $k < 0$ . Let $D_k = \pi/\sqrt{k}$ if $k > 0$ and $D_k = \infty$ if $k \le 0$ .

Let $(X, d)$ be a metric space. A geodesic segment in $X$ is a map $c \colon [0, \ell] \to X$ with $d(c(s), c(t)) = \abs{s - t}$ for all $s, t$ . A geodesic triangle consists of three points of $X$ and a geodesic segment between each pair of them. A comparison triangle for it is a triangle in $M_k$ with the same three side lengths. Each point $p$ on a side has a comparison point $\bar p$ on the matching side, at the same distances from the ends.

The space $X$ is a CAT( $k$ ) space if:

any two points of $X$ at distance less than $D_k$ are joined by a geodesic segment, and
for every geodesic triangle of perimeter less than $2 D_k$ and all points $p, q$ on its sides, $d(p, q) \le d_{M_k}(\bar p, \bar q)$ .

Remarks

Example: $\R^n$ with the Euclidean distance is CAT(0). Its geodesic triangles are flat triangles, and a comparison triangle in $M_0$ is congruent to it, so equality holds in clause 2.

Non-example: the unit round sphere $S^2$ is not CAT(0). Take the triangle with vertices on the equator at longitudes $0$ , $\pi/2$ and the north pole; its sides have length $\pi/2$ . The midpoints of the two sides through the pole are at spherical distance $\pi/3$ . The comparison triangle in $M_0$ is equilateral with side $\pi/2$ , so the comparison points are at distance $\pi/4 < \pi/3$ . So clause 2 fails; the sphere is CAT(1).

A CAT( $k$ ) space is also CAT( $k'$ ) for every $k' \ge k$ . A complete simply connected Riemannian manifold is CAT( $k$ ) for some $k \le 0$ exactly when its sectional curvature is at most $k$ . Metric trees are CAT( $k$ ) for every $k$ .

Tag	`KZ00AZ2HFM`
URI	`m2://KZ00AZ2HFM`
Kind	definition
Writing	W0 stub
MSC	53C23, 51F99
Defines	CAT(k) space
Uses	Metric space, Triangle, Euclidean space
Changed	2 November 2026

CAT(k) space

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