CAT (0) space

CAT (0) spaces are a geometry term used to generalize properties of manifolds of non-positive curvature to general metric spaces. Their defining property is that triangles should be thinner than comparison triangles in the Euclidean plane.

definition

Comparison triangles

${\displaystyle d(x,y)\leq \parallel x^{\prime }-y^{\prime }\parallel }$

Be a geodetic metric space . A geodesic triangle in is a triangle with corners whose three sides are geodesic. For every geodetic triangle there is a comparison triangle (which is unique apart from congruence ) in the with ${\displaystyle (X,d)}$ ${\displaystyle \Delta (a,b,c)}$ ${\displaystyle X}$ ${\displaystyle a,b,c\in X}$ ${\displaystyle \Delta (a,b,c)}$ ${\displaystyle \Delta (a^{\prime },b^{\prime },c^{\prime })}$ ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle d(a,b)=\lVert a^{\prime }-b^{\prime }\rVert ,\ d(a,c)=\lVert a^{\prime }-c^{\prime }\rVert ,\ d(b,c)=\lVert b^{\prime }-c^{\prime }\rVert }$.

You then have a comparison image

${\displaystyle f:\partial \Delta (a,b,c)\rightarrow \partial \Delta (a^{\prime },b^{\prime },c^{\prime })}$,

which (for example) assigns each point on the side the corresponding point on the side (i.e. the unique point with ), analogously for the other two sides. ${\displaystyle x}$ ${\displaystyle (a,b)}$ ${\displaystyle x^{\prime }}$ ${\displaystyle (a^{\prime },b^{\prime })}$ ${\displaystyle \lVert x^{\prime }-a^{\prime }\rVert =d(x,a)}$

CAT (0) spaces

A geodetic metric space is a CAT (0) -space if for every geodetic triangle in with comparison the inequality ${\displaystyle (X,d)}$ ${\displaystyle \Delta (a,b,c)}$ ${\displaystyle X}$ ${\displaystyle f:\partial \Delta (a,b,c)\rightarrow \partial \Delta (a^{\prime },b^{\prime },c^{\prime })}$

${\displaystyle d(x,y)\leq \lVert f(x)-f(y)\rVert }$applies to all . ${\displaystyle x,y\in \partial \Delta (a,b,c)}$

Clearly: every geodetic triangle is at least as thin as its comparison triangle.

Examples

• Simply connected complete Riemannian manifolds of non-positive section curvature are CAT (0) -spaces. These include the Euclidean , the hyperbolic space , more generally, all symmetric spaces without compact factor. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle SL(n,\mathbb {R} )/SO(n)}$
• Simply connected complete Riemannian manifolds of non-positive section curvature are also called Hadamard manifolds . Complete CAT (0) spaces are called Hadamard spaces .
• Finite products of CAT (0) spaces are CAT (0) spaces.
• ${\displaystyle \mathbb {R} }$Trees and Euclidean or hyperbolic buildings are CAT (0) spaces.
• Hilbert spaces are CAT (0) spaces.
• A contractible manifold of dimension carries a geodetically complete CAT (0) -metric if and only if it is collapsible . ${\displaystyle \geq 6}$
• Gromov's theorem : A cubic complex is a CAT (0) -space if and only if it is simply connected and the link of each vertex is a complex of plumes .

properties

• In a CAT (0) space , two points can be connected by a unique geodesic. The geodesic constantly depends on its endpoints. ${\displaystyle x}$
• The Ptolemaic inequality applies in CAT (0) spaces

${\displaystyle d(x,y)d(u,v)\leq d(x,u)d(y,v)+d(x,v)d(y,u)}$for everyone . ${\displaystyle x,y,u,v\in X}$

• For geodesics the function is convex . ${\displaystyle \gamma _{1},\gamma _{2}:\left[a,b\right]\rightarrow X}$ ${\displaystyle d(\gamma _{1}(t),\gamma _{2}(t))}$
• CAT (0) spaces are contractible .

Geodetic margin

Geodesic rays in a CAT (0) space are called asymptotic if they are finite apart. This defines an equivalence relation on the set of geodetic rays. The geodetic boundary of the CAT (0) space is the set of equivalence classes of geodetic rays parameterized to arc length. ${\displaystyle \partial _{\infty }X}$ ${\displaystyle (X,d)}$

Every point in can be connected to every point in by a unique geodesic. Different points in do not always have to be connected by a geodesic. ${\displaystyle \partial _{\infty }X}$ ${\displaystyle X}$ ${\displaystyle \partial _{\infty }X}$

Cone topology

The topology on can be expanded to a topology on , so that the following applies: A sequence converges to and only if (for anything ) the sequence of the and connecting geodesics locally evenly converges to the and connecting geodesics. ${\displaystyle (X,d)}$ ${\displaystyle X\cup \partial _{\infty }X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle p\in \partial _{\infty }X}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{n}}$ ${\displaystyle x_{0}}$ ${\displaystyle p}$

This topology is known as the cone topology .

Example: If a simply connected, complete n-dimensional Riemann manifold is non-positive sectional curvature, then with the cone topology it is homeomorphic to the (n-1) -dimensional sphere. ${\displaystyle (X,d)}$ ${\displaystyle \partial _{\infty }X}$

Tits metric

The Tits metric is defined by ${\displaystyle d_{T}:\partial _{\infty }X\times \partial _{\infty }X\rightarrow \mathbb {R} }$ ${\displaystyle p_{1},p_{2}\in \partial _{\infty }X}$

${\displaystyle d_{T}(p_{1},p_{2}):=\sup _{x\in X}\lim _{t\rightarrow \infty }\angle _{x}(\gamma _{1}(t),\gamma _{2}(t))}$,

where to are asymptotic geodesics. ${\displaystyle \gamma _{1},\gamma _{2}}$ ${\displaystyle p_{1},p_{2}}$

Here (generally for ) the angle is defined as the angle at the comparison triangle in . ${\displaystyle x,a,b\in X}$ ${\displaystyle \angle _{x}(a,b)}$ ${\displaystyle x^{\prime }}$ ${\displaystyle \Delta (x^{\prime },a^{\prime },b^{\prime })}$ ${\displaystyle \mathbb {R} ^{2}}$

The Tits metric does not generally induce the cone topology . ${\displaystyle \partial _{\infty }X}$

Examples: If a simply connected, complete Riemann manifold is of negative sectional curvature, then for all , the Tits metric induces the discrete topology . If is Euclidean space, then it is homeomorphic to the sphere. ${\displaystyle (X,d)}$ ${\displaystyle d_{T}(p_{1},p_{2})=\pi }$ ${\displaystyle p_{1},p_{2}\in \partial _{\infty }X}$ ${\displaystyle (X,d)=(\mathbb {R} ^{n},d_{eukl})}$ ${\displaystyle (\partial _{\infty }X,d_{T})}$

Horospheres

To a point , and a geodesic with one defines the Busemann function by ${\displaystyle p\in \partial _{\infty }X}$ ${\displaystyle \gamma :\left[0,\infty \right]\rightarrow X}$ ${\displaystyle \lim _{t\rightarrow \infty }\gamma (t)=p}$ ${\displaystyle b_{\gamma }:X\rightarrow \mathbb {R} }$

${\displaystyle b_{\gamma }(x):=\lim _{t\rightarrow \infty }d(x,\gamma (t))-t}$.

If is complete and and two are too asymptotic geodesics, then is constant. In particular, the decomposition of into the level sets of depends only on and not on the choice of the geodesic to be asymptotic . The levels of are called horospheres of . ${\displaystyle X}$ ${\displaystyle \gamma _{1}}$ ${\displaystyle \gamma _{2}}$ ${\displaystyle p\in \partial _{\infty }X}$ ${\displaystyle b_{\gamma _{1}}-b_{\gamma _{2}}}$ ${\displaystyle X}$ ${\displaystyle b_{\gamma }}$ ${\displaystyle p\in \partial _{\infty }X}$ ${\displaystyle p}$ ${\displaystyle \gamma }$ ${\displaystyle b_{\gamma }(x)}$ ${\displaystyle p}$

Isometrics

Every isometry of a complete CAT (0) space falls into one of the following 3 classes: ${\displaystyle f:X\rightarrow X}$ ${\displaystyle (X,d)}$

• elliptical :has a fixed point in, ${\displaystyle f}$ ${\displaystyle X}$
• hyperbolic :has no fixed point in, but leaves a geodesic invariant, ${\displaystyle f}$ ${\displaystyle X}$
• parabolic :leaves a pointand its horospheres invariant. ${\displaystyle f}$ ${\displaystyle p\in \partial _{\infty }X}$

CAT (0) groups

A CAT (0) group is a group that actually acts discontinuously and co-compactly through isometrics on a finite-dimensional CAT (0) space.

Local CAT (0) spaces

A complete, connected, metric space is locally called CAT (0) if every point has a neighborhood which (with the restricted metric) is a CAT (0) space.

A generalization of the Cartan-Hadamard theorem states that if there is a local CAT (0) space, then there is a unique metric on the universal overlay such that ${\displaystyle X}$ ${\displaystyle {\widetilde {X}}}$ ${\displaystyle {\tilde {d}}}$

• the overlay is a local isometric drawing , and ${\displaystyle {\widetilde {X}}\to X}$
• ${\displaystyle ({\widetilde {X}},{\tilde {d}})}$ is a CAT (0) space.

swell

1. ↑ Adiprasito-Funar: Hyperbolicity of contractible manifolds
2. ^ Bridson-Haefliger: Metric spaces of nonpositive curvature. (PDF file; 3.83 MB), definition II.8.5
3. ↑ Fujiwara: "CAT (0) spaces for Riemannian geometers" (PDF file; 116 kB)