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In mathematics, a CAT⁡(k){displaystyle mathbf {operatorname {textbf {CAT}} (k)} } space, where k{displaystyle k} is a real number, is a specific type of metric space. Intuitively, triangles in a CAT⁡(k){displaystyle operatorname {CAT} (k)} space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k{displaystyle k}. In a CAT⁡(k){displaystyle operatorname {CAT} (k)} space, the curvature is bounded from above by k{displaystyle k}. A notable special case is k=0{displaystyle k=0}; complete CAT⁡(0){displaystyle operatorname {CAT} (0)} spaces are known as Hadamard spaces after the French mathematician Jacques Hadamard.

Originally, Aleksandrov called these spaces “Rk{displaystyle {mathfrak {R}}_{k}} domain”.
The terminology CAT⁡(k){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.

For a real number k{displaystyle k}, let Mk{displaystyle M_{k}} denote the unique complete simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature k{displaystyle k}. Denote by Dk{displaystyle D_{k}} the diameter of Mk{displaystyle M_{k}}, which is +∞{displaystyle +infty } if k≤0{displaystyle kleq 0} and πk{displaystyle {frac {pi }{sqrt {k}}}} for k>0{displaystyle k>0}.

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

L(γ)=sup{∑i=1rd(γ(ti−1),γ(ti))|a=t0<t1<⋯<tr=b,r∈N}{displaystyle 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,rin mathbb {N} right}}

is precisely d(x,y){displaystyle d(x,y)}. Let Δ{displaystyle Delta } be a triangle in X{displaystyle X} with geodesic segments as its sides. Δ{displaystyle Delta } is said to satisfy the CAT⁡(k){displaystyle mathbf {operatorname {textbf {CAT}} (k)} } inequality if there is a comparison triangle Δ′{displaystyle Delta '} in the model space Mk{displaystyle M_{k}}, with sides of the same length as the sides of Δ{displaystyle Delta }, such that distances between points on Δ{displaystyle Delta } are less than or equal to the distances between corresponding points on Δ′{displaystyle Delta '}.

The geodesic metric space (X,d){displaystyle (X,d)} is said to be a CAT⁡(k){displaystyle mathbf {operatorname {textbf {CAT}} (k)} } space if every geodesic triangle Δ{displaystyle Delta } in X{displaystyle X} with perimeter less than 2Dk{displaystyle 2D_{k}} satisfies the CAT⁡(k){displaystyle operatorname {CAT} (k)} inequality. A (not-necessarily-geodesic) metric space (X,d){displaystyle (X,,d)} is said to be a space with curvature ≤k{displaystyle leq k} if every point of X{displaystyle X} has a geodesically convex CAT⁡(k){displaystyle operatorname {CAT} (k)} neighbourhood. A space with curvature ≤0{displaystyle leq 0} may be said to have non-positive curvature.

Examples

• Any CAT⁡(k){displaystyle operatorname {CAT} (k)} space (X,d){displaystyle (X,d)} is also a CAT⁡(ℓ){displaystyle operatorname {CAT} (ell )} space for all ℓ>k{displaystyle ell >k}. In fact, the converse holds: if (X,d){displaystyle (X,d)} is a CAT⁡(ℓ){displaystyle operatorname {CAT} (ell )} space for all ℓ>k{displaystyle ell >k}, then it is a CAT⁡(k){displaystyle operatorname {CAT} (k)} space.


• The n{displaystyle n}-dimensional Euclidean space En{displaystyle mathbf {E} ^{n}} with its usual metric is a CAT⁡(0){displaystyle operatorname {CAT} (0)} space. More generally, any real inner product space (not necessarily complete) is a CAT⁡(0){displaystyle operatorname {CAT} (0)} space; conversely, if a real normed vector space is a CAT⁡(k){displaystyle operatorname {CAT} (k)} space for some real k{displaystyle k}, then it is an inner product space.


• The n{displaystyle n}-dimensional hyperbolic space Hn{displaystyle mathbf {H} ^{n}} with its usual metric is a CAT⁡(−1){displaystyle operatorname {CAT} (-1)} space, and hence a CAT⁡(0){displaystyle operatorname {CAT} (0)} space as well.


• The n{displaystyle n}-dimensional unit sphere Sn{displaystyle mathbf {S} ^{n}} is a CAT⁡(1){displaystyle operatorname {CAT} (1)} space.


• More generally, the standard space Mk{displaystyle M_{k}} is a CAT⁡(k){displaystyle operatorname {CAT} (k)} space. So, for example, regardless of dimension, the sphere of radius r{displaystyle r} (and constant curvature 1r2{displaystyle {frac {1}{r^{2}}}}) is a CAT⁡(1r2){displaystyle operatorname {CAT} ({frac {1}{r^{2}}})} space. Note that the diameter of the sphere is πr{displaystyle pi r} (as measured on the surface of the sphere) not 2r{displaystyle 2r} (as measured by going through the centre of the sphere).


• The punctured plane Π=E2∖{0}{displaystyle Pi =mathbf {E} ^{2}backslash {mathbf {0} }} is not a CAT⁡(0){displaystyle operatorname {CAT} (0)} space since it is not geodesically convex (for example, the points (0,1){displaystyle (0,1)} and (0,−1){displaystyle (0,-1)} cannot be joined by a geodesic in Π{displaystyle Pi } with arc length 2), but every point of Π{displaystyle Pi } does have a CAT⁡(0){displaystyle operatorname {CAT} (0)} geodesically convex neighbourhood, so Π{displaystyle Pi } is a space of curvature ≤0{displaystyle leq 0}.


• The closed subspace X{displaystyle X} of E3{displaystyle mathbf {E} ^{3}} given by

X=E3∖{(x,y,z)|x>0,y>0 and z>0}{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 CAT⁡(k){displaystyle operatorname {CAT} (k)} space for any k{displaystyle k}.

• Any product of CAT⁡(0){displaystyle operatorname {CAT} (0)} spaces is CAT⁡(0){displaystyle operatorname {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 (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{displaystyle sigma _{1},sigma _{2}} are two geodesics in X defined on the same interval of time I, then the function I→R{displaystyle Ito mathbb {R} } given by

t↦d(σ1(t),σ2(t)){displaystyle tmapsto d{big (}sigma _{1}(t),sigma _{2}(t){big )}}

is convex in t.

Properties of CAT⁡(k){displaystyle operatorname {CAT} (k)} spaces

Let (X,d){displaystyle (X,d)} be a CAT⁡(k){displaystyle operatorname {CAT} (k)} space. Then the following properties hold:

• Given any two points x,y∈X{displaystyle x,yin X} (with d(x,y)<Dk{displaystyle d(x,y)<D_{k}} if k>0{displaystyle k>0}), there is a unique geodesic segment that joins x{displaystyle x} to y{displaystyle y}; moreover, this segment varies continuously as a function of its endpoints.


• Every local geodesic in X{displaystyle X} with length at most Dk{displaystyle D_{k}} is a geodesic.


• The d{displaystyle d}-balls in X{displaystyle X} of radius less than Dk/2{displaystyle D_{k}/2} are (geodesically) convex.


• The d{displaystyle d}-balls in X{displaystyle X} of radius less than Dk{displaystyle D_{k}} are contractible.


• Approximate midpoints are close to midpoints in the following sense: for every λ<Dk{displaystyle lambda <D_{k}} and every ϵ>0{displaystyle epsilon >0} there exists a δ=δ(k,λ,ϵ)>0{displaystyle delta =delta (k,lambda ,epsilon )>0} such that, if m{displaystyle m} is the midpoint of a geodesic segment from x{displaystyle x} to y{displaystyle y} with d(x,y)≤λ{displaystyle d(x,y)leq lambda } and

max{d(x,m′),d(y,m′)}≤12d(x,y)+δ,{displaystyle max {big {}d(x,m'),d(y,m'){big }}leq {frac {1}{2}}d(x,y)+delta ,}

then d(m,m′)<ϵ{displaystyle d(m,m')<epsilon }.

• It follows from these properties that, for k≤0{displaystyle kleq 0} the universal cover of every CAT⁡(k){displaystyle operatorname {CAT} (k)} space is contractible; in particular, the higher homotopy groups of such a space are trivial. As the example of the n{displaystyle n}-sphere Sn{displaystyle mathbf {S} ^{n}} shows, there is, in general, no hope for a CAT⁡(k){displaystyle operatorname {CAT} (k)} space to be contractible if k>0{displaystyle k>0}.


• An n{displaystyle n}-dimensional CAT⁡(k){displaystyle operatorname {CAT} (k)} space equipped with the n{displaystyle n}-dimensional Hausdorff measure satisfies the CD⁡[n,(n−1)k]{displaystyle operatorname {CD} [n,(n-1)k]} condition in the sense of John Lott, Cédric Villani, and Karl-Theodor Sturm.^{[citation needed]}
  

See also

• Cartan–Hadamard theorem

References

• 
  Alexander, S.; Kapovitch V.; Petrunin A. "Alexandrov Geometry, Chapter 7" (PDF). Retrieved 2011-04-07..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  Alexander, Stephanie; Kapovitch, Vitali; Petrunin, Anton. "Invitation to Alexandrov geometry: CAT[0] spaces". arXiv:1701.03483 [math.DG].
  


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  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.
  


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  Gromov, Mikhail (1987). "Hyperbolic groups". Essays in group theory. Math. Sci. Res. Inst. Publ. 8. New York: Springer. pp. 75–263. MR 0919829.
  


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  Hindawi, Mohamad A. (2005). Asymptotic invariants of Hadamard manifolds (PDF). University of Pennsylvania: PhD thesis.