View source for CAT(k) space

{{Short description|Type of metric space in mathematics}}
{{DISPLAYTITLE:CAT(''k'') space}}
In [[Mathematics|mathematics]], a <math>\mathbf{\operatorname{\textbf{CAT}}}(k)</math> '''space''', where <math>k</math> is a real number, is a specific type of [[Metric space|metric space]]. Intuitively, [[Triangle|triangle]]s in a <math>\operatorname{CAT}(k)</math> space (with <math>k<0</math>) are "slimmer" than corresponding "model triangles" in a standard space of [[Constant curvature|constant curvature]] <math>k</math>. In a <math>\operatorname{CAT}(k)</math> space, the curvature is bounded from above by <math>k</math>. A notable special case is <math>k=0</math>; complete <math>\operatorname{CAT}(0)</math> spaces are known as "[[Hadamard space]]s" after the {{wipe|France}} [[Biography:Mathematician|mathematician]] [[Biography:Jacques Hadamard|Jacques Hadamard]].

Originally, [[Biography:Aleksandr Danilovich Aleksandrov|Aleksandrov]] called these spaces “<math>\mathfrak{R}_k</math> domains”.
The terminology <math>\operatorname{CAT}(k)</math> was coined by Mikhail Gromov in 1987 and is an acronym for [[Biography:Élie Cartan|Élie Cartan]], [[Biography:Aleksandr Danilovich Aleksandrov|Aleksandr Danilovich Aleksandrov]] and [[Biography:Victor Andreevich Toponogov|Victor Andreevich Toponogov]] (although Toponogov never explored curvature bounded above in publications).

==Definitions==

[[Image:End of universe.jpg|thumb|Model triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.]]

For a [[Real number|real number]] <math>k</math>, let <math>M_k</math> denote the unique complete [[Simply connected space|simply connected]] [[Surface (mathematics)|surface]] (real 2-dimensional [[Riemannian manifold]]) with constant curvature <math>k</math>. Denote by <math>D_k</math> the [[Diameter|diameter]] of <math>M_k</math>, which is <math>\infty</math> if <math>k \leq 0</math> and is <math>\frac{\pi}{\sqrt{k}}</math> if <math>k>0</math>.

Let <math>(X,d)</math> be a geodesic metric space, i.e. a metric space for which every two points <math>x,y\in X</math> can be joined by a geodesic segment, an [[Arc length|arc length]] parametrized [[Curve|continuous curve]] <math>\gamma\colon [a,b] \to X,\ \gamma(a) = x,\ \gamma(b) = y</math>, whose length

:<math>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\}</math>

is precisely <math>d(x,y)</math>. Let <math>\Delta</math> be a triangle in <math>X</math> with geodesic segments as its sides. <math>\Delta</math> is said to satisfy the <math>\mathbf{\operatorname{\textbf{CAT}}}(k)</math> '''inequality''' if there is a [[Comparison triangle|comparison triangle]] <math>\Delta'</math> in the model space <math>M_k</math>, with sides of the same length as the sides of <math>\Delta</math>, such that distances between points on <math>\Delta</math> are less than or equal to the distances between corresponding points on <math>\Delta'</math>.

The geodesic metric space <math>(X,d)</math> is said to be a <math>\mathbf{\operatorname{\textbf{CAT}}}(k)</math> '''space''' if every geodesic triangle <math>\Delta</math> in <math>X</math> with [[Perimeter|perimeter]] less than <math>2D_k</math> satisfies the <math>\operatorname{CAT}(k)</math> inequality. A (not-necessarily-geodesic) metric space <math>(X,\,d)</math> is said to be a space with curvature <math>\leq k</math> if every point of <math>X</math> has a [[Geodesic convexity|geodesically convex]] <math>\operatorname{CAT}(k)</math> [[Neighbourhood (mathematics)|neighbourhood]]. A space with curvature <math>\leq 0</math> may be said to have [[Non-positive curvature|non-positive curvature]].

==Examples==

* Any <math>\operatorname{CAT}(k)</math> space <math>(X,d)</math> is also a <math>\operatorname{CAT}(\ell)</math> space for all <math>\ell>k</math>. In fact, the converse holds: if <math>(X,d)</math> is a <math>\operatorname{CAT}(\ell)</math> space for all <math>\ell>k</math>, then it is a <math>\operatorname{CAT}(k)</math> space.
* The <math>n</math>-dimensional [[Euclidean space]] <math>\mathbf{E}^n</math> with its usual metric is a <math>\operatorname{CAT}(0)</math> space. More generally, any real [[Inner product space|inner product space]] (not necessarily complete) is a <math>\operatorname{CAT}(0)</math> space; conversely, if a real [[Normed vector space|normed vector space]] is a <math>\operatorname{CAT}(k)</math> space for some real <math>k</math>, then it is an inner product space.
* The <math>n</math>-dimensional [[Hyperbolic space|hyperbolic space]] <math>\mathbf{H}^n</math> with its usual metric is a <math>\operatorname{CAT}(-1)</math> space, and hence a <math>\operatorname{CAT}(0)</math> space as well.
* The <math>n</math>-dimensional [[Unit sphere|unit sphere]] <math>\mathbf{S}^n</math> is a <math>\operatorname{CAT}(1)</math> space.
* More generally, the standard space <math>M_k</math> is a <math>\operatorname{CAT}(k)</math> space. So, for example, regardless of dimension, the sphere of radius <math>r</math> (and constant curvature <math display="inline">\frac{1}{r^2}</math>) is a <math display="inline">\operatorname{CAT}\left(\frac{1}{r^2}\right)</math> space. Note that the diameter of the sphere is <math>\pi r</math> (as measured on the surface of the sphere) not <math>2r</math> (as measured by going through the centre of the sphere).
* The punctured plane <math>\Pi = \mathbf{E}^2\backslash\{\mathbf{0}\}</math> is not a <math>\operatorname{CAT}(0)</math> space since it is not geodesically convex (for example, the points <math>(0,1)</math> and <math>(0,-1)</math> cannot be joined by a geodesic in <math>\Pi</math> with arc length 2), but every point of <math>\Pi</math> does have a <math>\operatorname{CAT}(0)</math> geodesically convex neighbourhood, so <math>\Pi</math> is a space of curvature <math>\leq 0</math>.
* The closed subspace <math>X</math> of <math>\mathbf{E}^3</math> given by <math>X = \mathbf{E}^{3} \setminus \{ (x, y, z) \mid x > 0, y > 0 \text{ and } z > 0 \}</math> equipped with the induced length metric is ''not'' a <math>\operatorname{CAT}(k)</math> space for any <math>k</math>.
* Any product of <math>\operatorname{CAT}(0)</math> spaces is <math>\operatorname{CAT}(0)</math>. (This does not hold for negative arguments.)

==Hadamard spaces==
{{main|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 manifold]]s. A Hadamard space is [[Contractible space|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 function|convex]]: if <math>\sigma_1, \sigma_2</math> are two geodesics in ''X'' defined on the same [[Interval (mathematics)|interval]] of time ''I'', then the function <math>I\to \R</math> given by

:<math>t \mapsto d \big( \sigma_{1} (t), \sigma_{2} (t) \big)</math>

is convex in ''t''.

==Properties of CAT(''k'') spaces==

Let <math>(X,d)</math> be a <math>\operatorname{CAT}(k)</math> space. Then the following properties hold:

* Given any two points <math>x,y\in X</math> (with <math>d(x,y)< D_k</math> if <math>k> 0</math>), there is a unique geodesic segment that joins <math>x</math> to <math>y</math>; moreover, this segment varies continuously as a function of its endpoints.
* Every local geodesic in <math>X</math> with length at most <math>D_k</math> is a geodesic.
* The <math>d</math>-balls in <math>X</math> of radius less than <math>D_k/2</math> are (geodesically) convex.
* The <math>d</math>-balls in <math>X</math> of radius less than <math>D_k</math> are contractible.
* Approximate midpoints are close to midpoints in the following sense: for every <math>\lambda < D_k</math> and every <math>\epsilon > 0</math> there exists a <math>\delta = \delta(k,\lambda,\epsilon) > 0</math> such that, if <math>m</math> is the midpoint of a geodesic segment from <math>x</math> to <math>y</math> with <math>d(x,y)\leq \lambda</math> and <math display="block">\max \bigl\{ d(x, m'), d(y, m') \bigr\} \leq \frac1{2} d(x, y) + \delta,</math> then <math>d(m,m') < \epsilon</math>.
* It follows from these properties that, for <math>k\leq 0</math> the universal cover of every <math>\operatorname{CAT}(k)</math> space is contractible; in particular, the higher [[Homotopy group|homotopy group]]s of such a space are [[Trivial group|trivial]].  As the example of the <math>n</math>-sphere <math>\mathbf{S}^n</math> shows, there is, in general, no hope for a <math>\operatorname{CAT}(k)</math> space to be contractible if <math>k > 0</math>.

==Surfaces of non-positive curvature==
In a region where the curvature of the surface satisfies {{math|''K'' ≤ 0}}, geodesic triangles satisfy the CAT(0) inequalities of '''comparison geometry''', studied by [[Biography:Élie Cartan|Cartan]], [[Biography:Aleksandr Danilovich Aleksandrov|Alexandrov]] and [[Biography:Victor Andreevich Toponogov|Toponogov]], and considered later from a different point of view by [[Biography:François Bruhat|Bruhat]] and [[Biography:Jacques Tits|Tits]].  Thanks to the vision of Gromov, this characterisation of non-positive curvature in terms of the underlying metric space has had a profound impact on modern geometry and in particular [[Geometric group theory|geometric group theory]]. Many results known for smooth surfaces and their geodesics, such as Birkhoff's method of constructing geodesics by his curve-shortening process or van Mangoldt and Hadamard's theorem that a simply connected surface of non-positive curvature is homeomorphic to the plane, are equally valid in this more general setting.

===Alexandrov's comparison inequality===
[[File:Median diagram.svg|thumb|right|The [[Median (geometry)|median]] in the comparison triangle is always longer than the actual median.]]
The simplest form of the comparison inequality, first proved for surfaces by Alexandrov around 1940, states that
{{quotation|The distance between a vertex of a geodesic triangle and the midpoint of the opposite side is always less than the corresponding distance in the comparison triangle in the plane with the same side-lengths.}}

The inequality follows from the fact that if {{math|''c''(''t'')}} describes a geodesic parametrized by arclength and {{math|''a''}} is a fixed point, then
:{{math|''f''(''t'') {{=}} ''d''(''a'',''c''(''t''))<sup>2</sup> − ''t''<sup>2</sup>}}
is a [[Convex function|convex function]], i.e.
:<math>\ddot{f}(t) \ge 0.</math>
Taking geodesic polar coordinates with origin at {{math|''a''}} so that {{math|‖''c''(''t'')‖ {{=}} ''r''(''t'')}}, convexity is equivalent to
:<math> r\ddot{r} + \dot{r}^2 \ge 1.</math>
Changing to normal coordinates {{math|''u''}}, {{math|''v''}} at {{math|''c''(''t'')}}, this inequality becomes
:{{math|''u''<sup>2</sup> + ''H''<sup>−1</sup>''H''<sub>r</sub>''v''<sup>2</sup> ≥ 1}},
where {{math|(''u'',''v'')}} corresponds to the unit vector {{math|''ċ''(''t'')}}. This follows from the inequality {{math|''H''<sub>''r''</sub> ≥ ''H''}}, a consequence of the non-negativity of the derivative of the [[Wronskian]] of {{math|''H''}} and {{math|''r''}} from [[Sturm–Liouville theory]].<ref name="Jost1997">{{harvnb|Berger|2004}}; {{citation|last=Jost|first=Jürgen|title= Nonpositive curvature: geometric and analytic aspects |publisher=Birkhäuser|year=1997|isbn=978-0-8176-5736-9|series=Lectures in Mathematics, ETH Zurich}}</ref>

== See also ==

* [[Cartan–Hadamard theorem]]

==References==
{{Reflist}}
* {{cite web
|last1= Alexander
|first1=Stephanie
|last2=Kapovitch
|first2=Vitali
|last3=Petrunin
|first3=Anton
| title = Alexandrov Geometry, Chapter 7
| url = http://www.math.uiuc.edu/~sba/the-defs-CBA.pdf
| format = PDF
| accessdate = 2011-04-07
}}
*{{cite arXiv |last1= Alexander|first1= Stephanie|last2=Kapovitch|first2=Vitali|last3=Petrunin|first3=Anton|eprint= 1701.03483|title= Invitation to Alexandrov geometry: CAT[0] spaces|class= math.DG}}
* {{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
| mr = 1377265
}}
* {{cite book |last1=Berger |first1=Marcel |title=A panoramic view of Riemannian geometry |date=2004 |publisher=Springer-Verlag |location=Berlin |isbn=978-3-540-65317-2}}
* {{cite book
| last1 = Bridson
| first1 = Martin R.
| last2=Haefliger
| first2=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
| mr = 1744486
}}
* {{cite book
| last = Gromov
| first = Mikhail
| chapter = Hyperbolic groups
| title = Essays in group theory
| series = Math. Sci. Res. Inst. Publ. 8
| pages = 75&ndash;263
| publisher = Springer
| location = New York
| year = 1987
| mr = 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
}}

[[Category:Metric geometry]]

{{Sourceattribution|CAT(k) space}}