Non-positively curved metric spaces

This post is based on parts of Martin Bridson’s 2006 ICM article.

In the next post we will give an overview of non-positively curved groups. In preparation, here is an introduction to non-positively curved metric spaces.

CAT(0) and CAT(-1) spaces

Named by Gromov in honour of Cartan, Alexandrov and Toponogov, a CAT({\kappa}) condition is a notion of curvature for a geodesic space — a metric space in which every pair of points {a}, {b} is connected by a path of length {d(a,b)}.

The CAT({\kappa}) condition compares geodesic triangles in {X} with triangles in a model space of constant curvature {\kappa}. It is a generalisation of a classical notion of curvature: assuming they are sufficiently smooth (such as {C^3}), then for Riemannian manifolds, locally the CAT({\kappa}) condition is equivalent to all the sectional curvatures being at most {\kappa} — see the Appendix to Chapter II.1 of Part II of Bridson and Haefliger’s book. We will limit our discussion to CAT(0) and CAT(-1) spaces; in these cases the model spaces are the Euclidean plane {\mathbb{E}^2} and the hyperbolic plane {\mathbb{H}^2}, respectively.

Here is what the CAT(0) and CAT(-1) conditions demand. Suppose {\Delta} is a geodesic triangle in {X}. Let {\bar{\Delta}} be a triangle (a comparison triangle) in {\mathbb{E}^2} (for the CAT(0) condition) or {\mathbb{H}^2} (for the stronger CAT(-1) condition) with the same side–lengths as {\Delta}. Suppose {p} and {q} are points on {\Delta}. Let {\bar{p}} and {\bar{q}} be the corresponding points on {\bar{\Delta}} — that is, the sides they lie on are those that correspond to the sides {p} and {q} lie on, and their locations on those side agree with those of {p} and {q} on theirs. Then {d(p,q) \leq d(\bar{p}, \bar{q})}.

comparison triangles

The usefulness of the CAT(0) and CAT(-1) conditions is greatly enhanced by the fact that they can readily be checked for suitable spaces (certain complexes) using Gromov’s link condition — more on which in some future post (probably).

We will now give two salient geometric features of a CAT(0) or CAT(-1) space {X}.

I. Triangles in CAT(-1) spaces are uniformly thin — that is, there exists {\delta \geq 0} such that every geodesic triangle in {X} is {\delta}–thin, meaning that each side is in the {\delta}-neighbourhood of the other two sides. This is true of CAT(-1) spaces because it holds for {\mathbb{H}^2}, where in fact {\delta} can be taken to be any number greater than {\ln(1+ \sqrt{2})}. As explained in Gersten’s Banff Notes, one needs only consider ideal triangles since a triangle’s vertices can be pushed out to the boundary whilst only making it less thin. Then, as ideal triangles in {\mathbb{H}^2} are equivalent, we can pick our favourite one; calculating {\delta} for that triangle is a straightforward calculus problem.

II. The metric in a CAT(0) space is convex — that is, if {c_1, c_2 : [0,1] \rightarrow X} are geodesics, parametrized proportional to arc–length, then

\displaystyle D(t) \ \leq \ (1-t) D(0) + tD(1)

for all {t \in [0,1]}, where {D(t) := d(c_1(t), c_2(t))}. Again, to see why this is true in {X}, we look at the corresponding claim in the model space — this time, {\mathbb{E}^2}. Suppose {\bar{c}_1, \bar{c}_2 : [0,1] \rightarrow \mathbb{E}^2} follow opposite sides of a quadrilateral and {\bar{c}_3 : [0,1] \rightarrow \mathbb{E}^2} follows the diagonal from {\bar{c}_2(0)} to {\bar{c}_1(1)} (all parametrized proportional to arc length). Then {\bar{D}(t) \leq (1-t) \bar{D}(0) + t\bar{D}(1)} for all {t \in [0,1]}, where {\bar{D}(t) := d(\bar{c}_1(t), \bar{c}_2(t))}, since {(1-t) \bar{D}(0)} is the distance from {\bar{c}_1(t)} to {\bar{c}_3(t)} and {t\bar{D}(1)} is that from {\bar{c}_3(t)} to {\bar{c}_2(t)}. So if {c_3} is a geodesic from {c_2(0)} to {c_1(1)}, then {d(c_1(t), c_3(t)) \leq d(\bar{c}_1(t), \bar{c}_3(t))} and {d(c_3(t), c_2(t)) \leq d(\bar{c}_3(t), \bar{c}_2(t))} by the CAT(0) condition, and therefore {D(t) \leq \bar{D}(t)}. So

\displaystyle D(t) \ \leq \ \bar{D}(t) \ \leq \ (1-t) \bar{D}(0) + t\bar{D}(1) \ = \ (1-t) D(0) + t D(1).


Convexity has two important consequences.

  1. Uniqueness of geodesics. For each pairs of points in {X}, there is a unique geodesic from one to the other.
  2. Contractibility. The space can be null–homotoped to a basepoint {x_0 \in X} by {h_t : X \rightarrow X} which takes {x} to the point on the geodesics from {x_0} to {x} a distance {(1-t)d(x_0,x)} from {x_0}. (Thus CAT(0) spaces have the potential to be universal covers of classifying spaces for discrete groups.)

About berstein

berstein is the name under which participants in the Berstein Seminar - a mathematics seminar at Cornell - are blogging.
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