# CAT(k) space

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 (with $k<0$) 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 “${\mathfrak {R}}_{k}$ domains”. 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

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 is ${\frac {\pi }{\sqrt {k}}}$ if $k>0$.
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
    $L(\gamma )=\sup \left\{\left.\sum _{i=1}^{r}d{\big\gamma (t_{i-1}),\gamma (t_{i}){\big )}\right|a=t_{0}<t_{1}<\cdots <t_{r}=b,r\in \mathbb {N} \right\}$
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 1/r$^2$ {\textstyle {\frac {1}{r^{2}}}}) is a CAT⁡(1/r$^2$){\textstyle \operatorname {CAT} \left({\frac {1}{r^{2}}}\right)} 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 $X=\mathbf {E} ^{3}\setminus \{(x,y,z)\mid x>0,y>0{\text{ and }}z>0\}$ 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.)

## Hadamard spaces

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 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 $d(x,y)<D_{k}$ 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 $\lambda <D_{k}$ 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 $\max {\bigl \d(x,m'),d(y,m'){\bigr \}}\leq {\frac {1}{2}}d(x,y)+\delta ,$ 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))$^2$ − t$^2$
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
    u$^2$ + H$^{−1}$H$_r$v$^2$ ≥ 1,
where (u,v) corresponds to the unit vector ċ(t). This follows from the inequality H$_r$ ≥ H, a consequence of the non-negativity of the derivative of the Wronskian of H and r from Sturm–Liouville theory.