CAT(k) space

In mathematics, a CAT("k") space is a specific type of metric space. Intuitively, triangles in a CAT("k") space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature "k". In a CAT("k") space, the curvature is bounded from above by "k". A notable special case is "k" = 0: complete CAT(0) spaces are known as Hadamard spaces after the French mathematician Jacques Hadamard. The terminology "CAT("k")" was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov.

Definitions

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 +∞ if "k" ≤ 0 and "π"/√"k" for "k" > 0.

Let ("X", "d") be a geodesic metric space, i.e. a metric space for which every two points "x", "y" ∈ "X" can be joined by a geodesic segment, an arc length parametrized continuous curve "γ" : ["a", "b"] → "X", "γ"("a") = "x", "γ"("b") = "y", whose length

:

is precisely "d"("x", "y"). Let Δ be a triangle in "X" with geodesic segments as its sides. Δ is said to satisfy the CAT("k") inequality if it has a comparison triangle Δ′ in the model space "M"_"k" and distances between points on Δ are less than or equal to the distances between corresponding points on Δ′.

The geodesic metric space ("X", "d") is said to be a CAT("k") space if every geodesic triangle Δ in "X" with perimeter less than 2"D"_"k" satisfies the CAT("k") inequality. A (not-necessarily-geodesic) metric space ("X", "d") is said to be a space with curvature ≤ "k" if every point of "X" has a geodesically convex CAT("k") neighbourhood. A space with curvature ≤ 0 may be said to have non-positive curvature.

Examples

* Any CAT("k") space ("X", "d") is also a CAT("ℓ") space for all "ℓ" > "k". In fact, the converse holds: if ("X", "d") is a CAT("ℓ") space for all "ℓ" > "k", then it is a CAT("k") space.
* "n"-dimensional Euclidean space E^"n" with its usual metric is a CAT(0) space. More generally, any real inner product space (not necessarily complete) is a CAT(0) space; conversely, if a real normed vector space is a CAT("k") space for some real "k", then it is an inner product space.
* "n"-dimensional hyperbolic space H^"n" with its usual metric is a CAT(−1) space, and hence a CAT(0) space as well.
* The "n"-dimensional unit sphere S^"n" is a CAT(1) space.
* More generally, the standard space "M"_"k" is a CAT("k") space. So, for example, regardless of dimension, the sphere of radius "r" (and constant curvature 1/√"r") is a CAT(1/√"r") space. Note that the diameter of the sphere is "πr" (as measured on the surface of the sphere) not 2"r" (as measured by going through the centre of the sphere).
* The punctured plane Π = E² {0} is not a 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 Π with arc length 2), but every point of Π does have a CAT(0) geodesically convex neighbourhood, so Π is a space of curvature ≤ 0.
* The closed subspace "X" of E³ given by

::$X\; =\; mathbf\{E\}^\{3\}\; setminus\; \{\; (x,\; y,\; z)\; |\; x\; >\; 0,\; y\; >\; 0\; mbox\{\; and\; \}\; z\; >\; 0\; \}$

:equipped with the induced length metric is "not" a CAT("k") space for any "k".
* Any product of CAT(0) spaces is 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. Most importantly, distance functions in Hadamard spaces are convex: if "σ"₁, "σ"₂ are two geodesics in "X" defined on the same interval of time "I", then the function "I" → R given by

:

is convex in "t".

Properties of CAT("k") spaces

Let ("X", "d") be a CAT("k") space. Then the following properties hold:

* Given any two points "x", "y" ∈ "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 ½"D"_"k" are (geodesically) convex.

* The "d"-balls in "X" of radius less than "D"_"k" are contractible.

* Approximate mid-points are close to mid-points in the following sense: for every "λ" < "D"_"k" and every "ε" > 0, there exists a "δ" = "δ"("k", "λ", "ε") > 0 such that, if "m" is the mid-point of a geodesic segment from "x" to "y" with "d"("x", "y") ≤ "λ" and

::

: then "d"("m", "m"′) < "ε".

* It follows from these properties that, for "k" ≤ 0, the universal cover of every CAT("k") space is contractible; in particular, the higher homotopy groups of such a space are trivial. As the example of the "n"-sphere S^"n" shows, there is, in general, no hope for a CAT("k") space to be contractible if "k" is strictly positive.

See also

* Cartan–Hadamard theorem

References

* cite book
last = Ballmann
first = Werner
title = Lectures on spaces of nonpositive curvature
series = DMV Seminar 25
publisher = Birkhäuser Verlag
location = Basel
year = 1995
pages = viii+112
isbn = 3-7643-5242-6 MathSciNet|id=1377265
* cite book
last = Bridson
first = Martin R.
coauthors = Haefliger, André
title = Metric spaces of non-positive curvature
series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319
publisher = Springer-Verlag
location = Berlin
year = 1999
pages = xxii+643
isbn = 3-540-64324-9 MathSciNet|id=1744486
* cite book
last = Gromov
first = Mikhail
authorlink = Mikhail Gromov
chapter = Hyperbolic groups
title = Essays in group theory
series = Math. Sci. Res. Inst. Publ. 8
pages = 75–263
publisher = Springer
location = New York
year = 1987 MathSciNet|id=919829
* cite book
last = Hindawi
first = Mohamad A.
title = Asymptotic invariants of Hadamard manifolds
publisher = PhD thesis
location = University of Pennsylvania
url = http://www.math.upenn.edu/grad/dissertations/HindawiThesis.pdf
year = 2005