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).

This theorem is very familiar to people working in coarse geometry, especially geometric group theorists. Because it is really a theorem in Riemannian geometry, rather than coarse geometry per se, its proof is often omitted in expositions of the theory; for example, I don’t believe there is a proof in Gromov-Ballmann-Schroeder or Ballmann (I think it is relegated to the exercises), nor is there a proof in Cheeger-Ebin, although one can piece together an argument from some of the ingredients in this last volume. Therefore I thought it might be a useful exercise to give a more-or-less complete exposition, which is reasonably self-contained and complete (Update: Daniel Groves tells me there is a proof in Bridson-Haefliger, which is good to know).

Part of what makes this a slightly fiddly theorem to prove is that one must somehow connect up the algebraic language of local Riemannian geometry with the metric language of distances, triangles, convexity and so on. The argument breaks up nicely into two parts — an infinitesimal comparison which is proved algebraically, and a global comparison which is derived from the local comparison by a “soft” argument. The first, algebraic part is not very deep, but it does contain an interesting nugget or two, which I will try to explain as I go along.

First, let’s briefly recall some of the ingredients of elementary Riemannian geometry. Given a Riemannian metric, there is a unique connection — the Levi-Civita connection — which is torsion-free, and compatible with the metric. We denote this by \nabla, so that \nabla_X Y denotes the covariant derivative of the vector field Y along the vector field X. For three vector fields X,Y,Z one defines the curvature tensor R(X,Y)Z:= \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z. Geometrically, this measures how Z rotates as one takes holonomy transport around an infinitesimal negatively oriented loop in the X-Y plane. The sectional curvature K in the X-Y plane is the ratio

K(X,Y):=\langle R(X,Y)Y,X\rangle/(\langle X,X\rangle\langle Y,Y\rangle - \langle X,Y\rangle^2)

The denominator of this expression is the area of a parallelogram spanned by X and Y, so if X,Y are orthogonal and of length 1, it reduces to 1.

If p is a point, and v \in T_pM is a tangent vector at that point, there is a unique geodesic \gamma:(-\epsilon,\epsilon) \to M with \gamma(0)=p and \gamma'(0) = v. If M is complete, \gamma(1) is defined; thus there is an exponential map from T_pM to M taking v to \gamma(1). If L is the subspace of T_pM spanned by a vector v, and u \in T_0T_pM, then we can define a vector field along L by setting it equal to s+tu at tv, for some constant s and for all t. The exponential map pushes this vector field forward to a vector field on M along \gamma, called a Jacobi field; by its construction, a Jacobi field is tangent to a 1-parameter variation of geodesics. A Jacobi field V satisfies the Jacobi equation \nabla_{\gamma'}\nabla_{\gamma'}V + R(V,\gamma')\gamma'=0. By abuse of notation, one identifies the frames along \gamma by parallel transport, and writes this as V'' + R(V,\gamma')\gamma'=0.

The easiest way to connect up the notions of curvature and comparison geometry is in the observation that for a manifold of nonpositive curvature, the norm of a Jacobi field is convex (as a function along a parameterized geodesic). We compute \gamma'\langle V,V'\rangle = \langle V',V'\rangle + \langle V,V''\rangle. Using the Jacobi equation, the second term can be rewritten, so this is equal to  \langle V',V'\rangle - \langle R(V,\gamma')\gamma',V\rangle. By the hypothesis that curvature is nonpositive, this is \ge \langle V',V'\rangle. We compute

|V|'' = \gamma'\left(\langle V,V'\rangle/|V|\right)=(\langle V',V'\rangle + \langle V,V''\rangle)/|V| - \langle V,V'\rangle^2/|V|^3

\ge \left(\langle V',V'\rangle\langle V,V\rangle - \langle V, V'\rangle^2 \right)/|V|^3 \ge 0

where the last inequality is just Cauchy-Schwarz.

OK, we are now ready to begin in earnest. Consider a geodesic \gamma from a to p, and a geodesic \delta through p making some angle \alpha with \gamma at p. Parameterize \delta by arc length s so that \delta(0)=p, and consider a 1-parameter family of geodesics \Gamma(s) from a to \delta(s). Note that \Gamma(0)=\gamma. If L(s) denotes the length of \Gamma(s), then the derivative dL/ds|_{s=0} = \cos(\alpha); in particular, it does not depend on the curvature of the space in question. The curvature manifests itself in second order information. The one-parameter family of geodesics \Gamma(s) is tangent along \gamma to a Jacobi field V, where V(a)=0 and V(p) = \delta'(0). Denote the vector field tangent to the \Gammas by T. The second variation formula (see e.g. Cheeger-Ebin pp. 20-21) says

d^2L/ds^2|_{s=0} = \langle \nabla_V V,T\rangle|^p_a + \int_a^p \langle V',V'\rangle - \langle R(V,T)T,V\rangle - (T\langle V,T\rangle)^2

Now, \nabla_V V vanishes at a, since V vanishes there; moreover at p it is tangent to \delta, and therefore vanishes there too. So the first term is zero. Furthermore, the term T\langle V,T\rangle = \langle V',T\rangle (since \nabla_T T = 0 because T is tangent to geodesics) and

T\langle V,T\rangle = \langle V'',T\rangle = -\langle R(V,T)T,T\rangle = 0

along \gamma, by the Jacobi equation applied to V. Hence T\langle V,T\rangle is constant along \gamma, and one sees that it contributes a term which depends only on the angle \alpha. Lets abbreviate I(V,V):=\int_a^p \langle V',V'\rangle - \langle R(V,T)T,V\rangle. Another simple calculation (see Cheeger-Ebin pp.24-25) shows that if W=fV for any function f with f(a)=f(p)=1 then I(W,W) \ge I(V,V); this is one of the fundamental (and standard) index lemmas, which say that in a suitable sense, Jacobi fields minimize the form I.

We are now ready to compare second derivatives in M and in our comparison space M_0. Let \gamma_0 and \delta_0 be geodesics as above in a comparison space of constant curvature K_0 with the same lengths as \gamma,\delta and making the same angle \alpha at their intersection. Let \Gamma_0 be the analogous 1-parameter family of geodesics, and let L_0(s) denote the length of \Gamma_0(s). We know that the first derivatives of L and L_0 agree, and would like to compare second derivatives. Apart from the term that depends only on the angle, this means comparing I(V,V) and I(V_0,V_0). This is basically a special case of the Rauch comparison theorem, and our argument is a simplification of Rauch. Let’s suppose for simplicity that both M and M_0 are 2-dimensional. Parallel transport along \gamma and \gamma_0 identifies the tangent spaces along these geodesics with the tangent spaces at \gamma(a) and \gamma_0(\bar{a}) respectively. Choosing an isometry between these tangent spaces which takes \gamma'(a) to \gamma_0'(\bar{a}), we can define the “pushforward” \tilde{V} to be a vector field along \gamma_0 satisfying \tilde{V}(p) = V_0(p) and \tilde{V}' = V'. By construction we can write \tilde{V} = fV_0 + W where W is tangent to T_0, and where f(\bar{a})=f(\bar{p}) = 1. Thus I(\tilde{V},\tilde{V})\ge I(V_0,V_0). On the other hand, \langle V',V'\rangle = \langle \tilde{V}',\tilde{V}'\rangle at comparable points by definition, and

-\langle R(V,T)T,V\rangle \ge -\langle R(\tilde{V},T_0)T_0,\tilde{V}\rangle

pointwise by the hypothesis comparing the curvature of M and M_0. Hence

I(V,V) \ge I(\tilde{V},\tilde{V}) \ge I(V_0,V_0)

and we conclude that the distance function to geodesics is more convex in M than in the comparison space M_0. This is the desired infinitesimal comparison theorem; it remains to bootstrap it to a global comparison theorem.

Right; let’s look at our comparison triangles abc and \bar{a}\bar{b}\bar{c}. By the hypothesis that M is simply-connected, we can actually map a disk into M spanning the geodesic triangle; a minimal area such disk will have intrinsic curvature bounded above by that of M, and distances in this disk between points on the boundary will be at least as large as they are in M. So without loss of generality, we may assume that M is 2-dimensional, and that abc is spanned by an honest triangular disk. Parameterize the side bc by length, and let p(t) be the point on bc with d_M(b,p(t))=t. Let \bar{p}(t) be the analogous point on \bar{b}\bar{c}. Define

f(t):=d_M(a,p(t)) and f_0(t):=d_{M_0}(\bar{a},\bar{p}(t)).

We know f(0)=f_0(0) = ab and f(bc)=f_0(bc) = ac. We would like to show f \le f_0 pointwise. Suppose not, and restrict to a maximal connected interval on which this fails. By the infinitesimal comparison theorem  proved above, this interval must have nonempty interior. Let b',c' and \bar{b}',\bar{c}' be the points on bc and \bar{b}\bar{c} corresponding to the endpoints of the interval. Evidently the triangles ab'c' and a'\bar{b}'\bar{c}' are also comparison triangles; so WLOG we may just take b'=b, c'=c and so on.

We now employ a trick. Consider a 1-parameter family of comparison triangles in spaces of constant curvature K(u)=K_0+u. The CAT(K) Theorem for spaces of constant curvature reduces to an explicit calculation, since the function L as above can be computed exactly, and we suppose the theorem proved for such spaces. It follows that as u increases, the function f_u(t) also increases monotonically. By assumption, for small u there is some t with f_u(t) < f(t). Eventually therefore we get some u and some intermediate t where f_u(t) = f(t) and f_u \ge f for all points near t. But this contradicts the infinitesimal comparison theorem proved above. qed.

The figure above illustrates the meaning of the last step. The blue curve is the graph of f, and the red curves are the graphs of f_u for various u. As u is increased, the red curves move upward in a family. There is some biggest u for which the red curve is not entirely above the blue curve, and for that curve, the red and blue curves have a point of tangency. But at that point of tangency we would have f_u''\ge f'', contrary to the infinitesimal comparison theorem which shows f'' \le f_0'' with equality iff the curvatures along the corresponding geodesics are pointwise equal, which they are not for u>0.