## 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 ${d(p,q) \leq d(\bar{p}, \bar{q})}$.

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.) 