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I am currently teaching a class at the University of Chicago on hyperbolic groups, and I have just introduced the concept of \delta-hyperbolic (geodesic) metric spaces. A geodesic metrix space (X,d_X) is \delta-hyperbolic if for any geodesic triangle abc, and any p \in ab there is some q \in ac \cup bc with d_X(p,q)\le \delta. The quintessential \delta-hyperbolic space is the hyperbolic plane, the unique (up to isometry) simply-connected complete Riemannian 2-manifold of constant curvature -1. It follows that any simply-connected complete Riemannian manifold of constant curvature K<0 is \delta-hyperbolic for some \delta depending on K; roughly one can take \delta \sim (-K)^{-1/2}.

What gives this condition some power is the rich class of examples of spaces which are \delta-hyperbolic for some \delta. One very important class of examples are simply-connected complete Riemannian manifolds with upper curvature bounds. Such spaces enjoy a very strong comparison property with simply-connected spaces of constant curvature, and are therefore the prime examples of what are known as CAT(K) spaces.

Definition: A geodesic metric space (X,d_X) is said to be CAT(K), if the following holds. If abc is a geodesic triangle in X, let \bar{a}\bar{b}\bar{c} be a comparison triangle in a simply connected complete Riemannian manifold Y of constant curvature K. Being a comparison triangle means just that the length of \bar{a}\bar{b} is equal to the length of ab and so on. For any p \in bc there is a corresponding point \bar{p} in the comparison edge \bar{b}\bar{c} which is the same distance from \bar{b} and \bar{c} as p is from b and c respectively. The CAT(K) condition says, for all abc as above, and all p \in bc, there is an inequality d_X(a,p) \le d_Y(\bar{a},\bar{p}).

The term CAT here (coined by Gromov) is an acronym for Cartan-Alexandrov-Toponogov, who all proved significant theorems in Riemannian comparison geometry. From the definition it follows immediately that any CAT(K) space with K<0 is \delta-hyperbolic for some \delta depending only on K. The point of this post is to give a short proof of the following fundamental fact:

CAT(K) Theorem: Let M be a complete simply-connected Riemannian manifold with sectional curvature \le K_0 everywhere. Then M with its induced Riemannian (path) metric is CAT(K_0).

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