# CAT(k) space

(Redirected from CAT(0))

In mathematics, a $\mathbf {\operatorname {\textbf {CAT}} (k)}$ space, where $k$ is a real number, is a specific type of metric space. Intuitively, triangles in a $\operatorname {CAT} (k)$ space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature $k$ . In a $\operatorname {CAT} (k)$ space, the curvature is bounded from above by $k$ . A notable special case is $k=0$ ; complete $\operatorname {CAT} (0)$ spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard.

Originally, Aleksandrov called these spaces “${\mathfrak {R}}_{k}$ domain”. The terminology $\operatorname {CAT} (k)$ was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications).

## Definitions

Model triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.

For a real number $k$ , let $M_{k}$  denote the unique complete simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature $k$ . Denote by $D_{k}$  the diameter of $M_{k}$ , which is $\infty$  if $k\leq 0$  and ${\frac {\pi }{\sqrt {k}}}$  for $k>0$ .

Let $(X,d)$  be a geodesic metric space, i.e. a metric space for which every two points $x,y\in X$  can be joined by a geodesic segment, an arc length parametrized continuous curve $\gamma \colon [a,b]\to X,\ \gamma (a)=x,\ \gamma (b)=y$ , whose length

$L(\gamma )=\sup \left\{\left.\sum _{i=1}^{r}d{\big (}\gamma (t_{i-1}),\gamma (t_{i}){\big )}\right|a=t_{0}

is precisely $d(x,y)$ . Let $\Delta$  be a triangle in $X$  with geodesic segments as its sides. $\Delta$  is said to satisfy the $\mathbf {\operatorname {\textbf {CAT}} (k)}$  inequality if there is a comparison triangle $\Delta '$  in the model space $M_{k}$ , with sides of the same length as the sides of $\Delta$ , such that distances between points on $\Delta$  are less than or equal to the distances between corresponding points on $\Delta '$ .

The geodesic metric space $(X,d)$  is said to be a $\mathbf {\operatorname {\textbf {CAT}} (k)}$  space if every geodesic triangle $\Delta$  in $X$  with perimeter less than $2D_{k}$  satisfies the $\operatorname {CAT} (k)$  inequality. A (not-necessarily-geodesic) metric space $(X,\,d)$  is said to be a space with curvature $\leq k$  if every point of $X$  has a geodesically convex $\operatorname {CAT} (k)$  neighbourhood. A space with curvature $\leq 0$  may be said to have non-positive curvature.

## Examples

• Any $\operatorname {CAT} (k)$  space $(X,d)$  is also a $\operatorname {CAT} (\ell )$  space for all $\ell >k$ . In fact, the converse holds: if $(X,d)$  is a $\operatorname {CAT} (\ell )$  space for all $\ell >k$ , then it is a $\operatorname {CAT} (k)$  space.
• The $n$ -dimensional Euclidean space $\mathbf {E} ^{n}$  with its usual metric is a $\operatorname {CAT} (0)$  space. More generally, any real inner product space (not necessarily complete) is a $\operatorname {CAT} (0)$  space; conversely, if a real normed vector space is a $\operatorname {CAT} (k)$  space for some real $k$ , then it is an inner product space.
• The $n$ -dimensional hyperbolic space $\mathbf {H} ^{n}$  with its usual metric is a $\operatorname {CAT} (-1)$  space, and hence a $\operatorname {CAT} (0)$  space as well.
• The $n$ -dimensional unit sphere $\mathbf {S} ^{n}$  is a $\operatorname {CAT} (1)$  space.
• More generally, the standard space $M_{k}$  is a $\operatorname {CAT} (k)$  space. So, for example, regardless of dimension, the sphere of radius $r$  (and constant curvature ${\frac {1}{r^{2}}}$ ) is a $\operatorname {CAT} ({\frac {1}{r^{2}}})$  space. Note that the diameter of the sphere is $\pi r$  (as measured on the surface of the sphere) not $2r$  (as measured by going through the centre of the sphere).
• The punctured plane $\Pi =\mathbf {E} ^{2}\backslash \{\mathbf {0} \}$  is not a $\operatorname {CAT} (0)$  space since it is not geodesically convex (for example, the points $(0,1)$  and $(0,-1)$  cannot be joined by a geodesic in $\Pi$  with arc length 2), but every point of $\Pi$  does have a $\operatorname {CAT} (0)$  geodesically convex neighbourhood, so $\Pi$  is a space of curvature $\leq 0$ .
• The closed subspace $X$  of $\mathbf {E} ^{3}$  given by
$X=\mathbf {E} ^{3}\setminus \{(x,y,z)|x>0,y>0{\text{ and }}z>0\}$
equipped with the induced length metric is not a $\operatorname {CAT} (k)$  space for any $k$ .
• Any product of $\operatorname {CAT} (0)$  spaces is $\operatorname {CAT} (0)$ . (This does not hold for negative arguments.)

As a special case, a complete CAT(0) space is also known as a Hadamard space; this is by analogy with the situation for Hadamard manifolds. A Hadamard space is contractible (it has the homotopy type of a single point) and, between any two points of a Hadamard space, there is a unique geodesic segment connecting them (in fact, both properties also hold for general, possibly incomplete, CAT(0) spaces). Most importantly, distance functions in Hadamard spaces are convex: if $\sigma _{1},\sigma _{2}$  are two geodesics in X defined on the same interval of time I, then the function $I\to \mathbb {R}$  given by

$t\mapsto d{\big (}\sigma _{1}(t),\sigma _{2}(t){\big )}$

is convex in t.

## Properties of $\operatorname {CAT} (k)$ spaces

Let $(X,d)$  be a $\operatorname {CAT} (k)$  space. Then the following properties hold:

• Given any two points $x,y\in X$  (with $d(x,y)  if $k>0$ ), there is a unique geodesic segment that joins $x$  to $y$ ; moreover, this segment varies continuously as a function of its endpoints.
• Every local geodesic in $X$  with length at most $D_{k}$  is a geodesic.
• The $d$ -balls in $X$  of radius less than $D_{k}/2$  are (geodesically) convex.
• The $d$ -balls in $X$  of radius less than $D_{k}$  are contractible.
• Approximate midpoints are close to midpoints in the following sense: for every $\lambda   and every $\epsilon >0$  there exists a $\delta =\delta (k,\lambda ,\epsilon )>0$  such that, if $m$  is the midpoint of a geodesic segment from $x$  to $y$  with $d(x,y)\leq \lambda$  and
$\max {\big \{}d(x,m'),d(y,m'){\big \}}\leq {\frac {1}{2}}d(x,y)+\delta ,$
then $d(m,m')<\epsilon$ .
• It follows from these properties that, for $k\leq 0$  the universal cover of every $\operatorname {CAT} (k)$  space is contractible; in particular, the higher homotopy groups of such a space are trivial. As the example of the $n$ -sphere $\mathbf {S} ^{n}$  shows, there is, in general, no hope for a $\operatorname {CAT} (k)$  space to be contractible if $k>0$ .

## Surfaces of non-positive curvature

In a region where the curvature of the surface satisfies K ≤ 0, geodesic triangles satisfy the CAT(0) inequalities of comparison geometry, studied by Cartan, Alexandrov and Toponogov, and considered later from a different point of view by Bruhat and Tits; thanks to the vision of Gromov, this characterisation of non-positive curvature in terms of the underlying metric space has had a profound impact on modern geometry and in particular geometric group theory. Many results known for smooth surfaces and their geodesics, such as Birkhoff's method of constructing geodesics by his curve-shortening process or van Mangoldt and Hadamard's theorem that a simply connected surface of non-positive curvature is homeomorphic to the plane, are equally valid in this more general setting.

### Alexandrov's comparison inequality

The simplest form of the comparison inequality, first proved for surfaces by Alexandrov around 1940, states that

The distance between a vertex of a geodesic triangle and the midpoint of the opposite side is always less than the corresponding distance in the comparison triangle in the plane with the same side-lengths.

The inequality follows from the fact that if c(t) describes a geodesic parametrized by arclength and a is a fixed point, then

f(t) = d(a,c(t))2t2

is a convex function, i.e.

${\ddot {f}}(t)\geq 0.$

Taking geodesic polar coordinates with origin at a so that c(t)‖ = r(t), convexity is equivalent to

$r{\ddot {r}}+{\dot {r}}^{2}\geq 1.$

Changing to normal coordinates u, v at c(t), this inequality becomes

u2 + H−1Hrv2 ≥ 1,

where (u,v) corresponds to the unit vector ċ(t). This follows from the inequality HrH, a consequence of the non-negativity of the derivative of the Wronskian of H and r from Sturm–Liouville theory.