# CAT(k) space

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

Originally, Aleksandrov called these spaces “${\displaystyle {\mathfrak {R}}_{k}}$ domain”. The terminology ${\displaystyle \operatorname {CAT} (k)}$ 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 ${\displaystyle k}$ , let ${\displaystyle M_{k}}$  denote the unique complete simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature ${\displaystyle k}$ . Denote by ${\displaystyle D_{k}}$  the diameter of ${\displaystyle M_{k}}$ , which is ${\displaystyle +\infty }$  if ${\displaystyle k\leq 0}$  and ${\displaystyle {\frac {\pi }{\sqrt {k}}}}$  for ${\displaystyle k>0}$ .

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

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

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

## Examples

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

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

is convex in t.

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

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

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