# 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 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, Alexandrov 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 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 $\gamma\,:\,[a,b] \to X,\ \gamma(a) = x,\ \gamma(b) = y$, 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.
• $n$-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.
• $n$-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 σ1, σ2 are two geodesics in X defined on the same interval of time I, then the function I → 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 $d(x,y)< D_k$ 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 $\frac{1}{2}D_k$ 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
$\max \big\{ d(x, m'), d(y, m') \big\} \leq \frac1{2} d(x, y) + \delta,$
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 $k > 0$.
• An $n$-dimensional $\operatorname{CAT}(k)$ space equipped with the $n$-dimensional Hausdorff measure satisfies the $\operatorname{CD}[n, (n-1)k]$ condition in the sense of Lott-Villani-Sturm[citation needed].