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  "body": "\n## Definition\n\nLet $k$ be a real number.\nLet $M_k$ be the complete simply connected surface of constant curvature $k$.\nIt is the round sphere of radius $1/\\sqrt{k}$ if $k \u003e 0$, the [[Euclidean plane|0XPR1E85W0]] if $k = 0$, and the hyperbolic plane of curvature $k$ if $k \u003c 0$.\nLet $D_k = \\pi/\\sqrt{k}$ if $k \u003e 0$ and $D_k = \\infty$ if $k \\le 0$.\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]].\nA *geodesic segment* in $X$ is a map $c \\colon [0, \\ell] \\to X$ with $d(c(s), c(t)) = \\abs{s - t}$ for all $s, t$.\nA *geodesic triangle* consists of three points of $X$ and a geodesic segment between each pair of them.\nA *comparison triangle* for it is a [[triangle|0CTNFGYMM2]] in $M_k$ with the same three side lengths.\nEach point $p$ on a side has a comparison point $\\bar p$ on the matching side, at the same distances from the ends.\n\nThe space $X$ is a *CAT($k$) space* if:\n\n1. any two points of $X$ at distance less than $D_k$ are joined by a geodesic segment, and\n2. for every geodesic triangle of perimeter less than $2 D_k$ and all points $p, q$ on its sides, $d(p, q) \\le d_{M_k}(\\bar p, \\bar q)$.\n\n## Remarks\n\nExample: $\\R^n$ with the Euclidean distance is CAT(0).\nIts geodesic triangles are flat triangles, and a comparison triangle in $M_0$ is congruent to it, so equality holds in clause 2.\n\nNon-example: the unit round sphere $S^2$ is not CAT(0).\nTake the triangle with vertices on the equator at longitudes $0$, $\\pi/2$ and the north pole; its sides have length $\\pi/2$.\nThe midpoints of the two sides through the pole are at spherical distance $\\pi/3$.\nThe comparison triangle in $M_0$ is equilateral with side $\\pi/2$, so the comparison points are at distance $\\pi/4 \u003c \\pi/3$.\nSo clause 2 fails; the sphere is CAT(1).\n\nA CAT($k$) space is also CAT($k'$) for every $k' \\ge k$.\nA complete simply connected Riemannian manifold is CAT($k$) for some $k \\le 0$ exactly when its [[sectional curvature|1X42JAA349]] is at most $k$.\nMetric trees are CAT($k$) for every $k$.\n",
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