I am currently teaching a class at the University of Chicago on hyperbolic groups, and I have just introduced the concept of -hyperbolic (geodesic) metric spaces. A geodesic metrix space is -hyperbolic if for any geodesic triangle , and any there is some with . The quintessential -hyperbolic space is the hyperbolic plane, the unique (up to isometry) simply-connected complete Riemannian 2-manifold of constant curvature . It follows that any simply-connected complete Riemannian manifold of constant curvature is -hyperbolic for some depending on ; roughly one can take .
What gives this condition some power is the rich class of examples of spaces which are -hyperbolic for some . 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 is said to be , if the following holds. If is a geodesic triangle in , let be a comparison triangle in a simply connected complete Riemannian manifold of constant curvature . Being a comparison triangle means just that the length of is equal to the length of and so on. For any there is a corresponding point in the comparison edge which is the same distance from and as is from and respectively. The condition says, for all as above, and all , there is an inequality .
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 space with is -hyperbolic for some depending only on . The point of this post is to give a short proof of the following fundamental fact:
CAT(K) Theorem: Let be a complete simply-connected Riemannian manifold with sectional curvature everywhere. Then with its induced Riemannian (path) metric is .
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 , so that denotes the covariant derivative of the vector field along the vector field . For three vector fields one defines the curvature tensor . Geometrically, this measures how rotates as one takes holonomy transport around an infinitesimal negatively oriented loop in the - plane. The sectional curvature in the - plane is the ratio
The denominator of this expression is the area of a parallelogram spanned by and , so if are orthogonal and of length 1, it reduces to 1.
If is a point, and is a tangent vector at that point, there is a unique geodesic with and . If is complete, is defined; thus there is an exponential map from to taking to . If is the subspace of spanned by a vector , and , then we can define a vector field along by setting it equal to at , for some constant and for all . The exponential map pushes this vector field forward to a vector field on along , called a Jacobi field; by its construction, a Jacobi field is tangent to a 1-parameter variation of geodesics. A Jacobi field satisfies the Jacobi equation . By abuse of notation, one identifies the frames along by parallel transport, and writes this as .
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 . Using the Jacobi equation, the second term can be rewritten, so this is equal to . By the hypothesis that curvature is nonpositive, this is . We compute
where the last inequality is just Cauchy-Schwarz.
OK, we are now ready to begin in earnest. Consider a geodesic from to , and a geodesic through making some angle with at . Parameterize by arc length so that , and consider a 1-parameter family of geodesics from to . Note that . If denotes the length of , then the derivative ; 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 is tangent along to a Jacobi field , where and . Denote the vector field tangent to the s by . The second variation formula (see e.g. Cheeger-Ebin pp. 20-21) says
Now, vanishes at , since vanishes there; moreover at it is tangent to , and therefore vanishes there too. So the first term is zero. Furthermore, the term (since because is tangent to geodesics) and
along , by the Jacobi equation applied to . Hence is constant along , and one sees that it contributes a term which depends only on the angle . Lets abbreviate . Another simple calculation (see Cheeger-Ebin pp.24-25) shows that if for any function with then ; this is one of the fundamental (and standard) index lemmas, which say that in a suitable sense, Jacobi fields minimize the form .
We are now ready to compare second derivatives in and in our comparison space . Let and be geodesics as above in a comparison space of constant curvature with the same lengths as and making the same angle at their intersection. Let be the analogous 1-parameter family of geodesics, and let denote the length of . We know that the first derivatives of and agree, and would like to compare second derivatives. Apart from the term that depends only on the angle, this means comparing and . 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 and are 2-dimensional. Parallel transport along and identifies the tangent spaces along these geodesics with the tangent spaces at and respectively. Choosing an isometry between these tangent spaces which takes to , we can define the “pushforward” to be a vector field along satisfying and . By construction we can write where is tangent to , and where . Thus . On the other hand, at comparable points by definition, and
pointwise by the hypothesis comparing the curvature of and . Hence
and we conclude that the distance function to geodesics is more convex in than in the comparison space . 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 and . By the hypothesis that is simply-connected, we can actually map a disk into spanning the geodesic triangle; a minimal area such disk will have intrinsic curvature bounded above by that of , and distances in this disk between points on the boundary will be at least as large as they are in . So without loss of generality, we may assume that is 2-dimensional, and that is spanned by an honest triangular disk. Parameterize the side by length, and let be the point on with . Let be the analogous point on . Define
We know and . We would like to show 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 and be the points on and corresponding to the endpoints of the interval. Evidently the triangles and are also comparison triangles; so WLOG we may just take , and so on.
We now employ a trick. Consider a 1-parameter family of comparison triangles in spaces of constant curvature . The CAT(K) Theorem for spaces of constant curvature reduces to an explicit calculation, since the function as above can be computed exactly, and we suppose the theorem proved for such spaces. It follows that as increases, the function also increases monotonically. By assumption, for small there is some with . Eventually therefore we get some and some intermediate where and for all points near . 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 , and the red curves are the graphs of for various . As is increased, the red curves move upward in a family. There is some biggest 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 , contrary to the infinitesimal comparison theorem which shows with equality iff the curvatures along the corresponding geodesics are pointwise equal, which they are not for .