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.)

Hadamard spaces

Main article: Hadamard space

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 $\lambda<D_k$ 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 \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 .

References

Alexander, S.; Kapovitch V.; Petrunin A. "Alexandrov Geometry, Chapter 7" (PDF). Retrieved 2011-04-07.
Ballmann, Werner (1995). Lectures on spaces of nonpositive curvature. DMV Seminar 25. Basel: Birkhäuser Verlag. pp. viii+112. ISBN 3-7643-5242-6. MR 1377265.
Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319. Berlin: Springer-Verlag. pp. xxii+643. ISBN 3-540-64324-9. MR 1744486.
Gromov, Mikhail (1987). "Hyperbolic groups". Essays in group theory. Math. Sci. Res. Inst. Publ. 8. New York: Springer. pp. 75–263. MR 919829.
Hindawi, Mohamad A. (2005). Asymptotic invariants of Hadamard manifolds. University of Pennsylvania: PhD thesis.