# 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, Alexandrov 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.

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 ${\displaystyle \gamma \,:\,[a,b]\to X,\ \gamma (a)=x,\ \gamma (b)=y}$, 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

${\displaystyle X={\mathbf {E} }^{3}\setminus \{(x,y,z)|x>0,y>0{\text{ and }}z>0\}}$
equipped with the induced length metric is not a ${\displaystyle \operatorname {CAT} (k)}$ space for any ${\displaystyle k}$.

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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

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

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:

${\displaystyle \max {\big \{}d(x,m'),d(y,m'){\big \}}\leq {\frac {1}{2}}d(x,y)+\delta ,}$
then ${\displaystyle d(m,m')<\epsilon }$.

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## References

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