In mathematics, a
CAT
(
k
)
space, where
k
is a real number, 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.
Originally, Alexandrov called these spaces “
R
k
domain”. 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 (although Toponogov never explored curvature bounded above in publications).
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
L
(
γ
)
=
sup
{
∑
i
=
1
r
d
(
γ
(
t
i
−
1
)
,
γ
(
t
i
)
)
|
a
=
t
0
<
t
1
<
⋯
<
t
r
=
b
,
r
∈
N
}
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 there is a comparison triangle
Δ
′
in the model space
M
k
, with sides of the same length as the sides of
Δ
, such that 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.
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
2
) is a
CAT
(
1
r
2
)
space. Note that the diameter of the sphere is
π
r
(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
=
E
2
∖
{
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
E
3
given by
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.)
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
↦
d
(
σ
1
(
t
)
,
σ
2
(
t
)
)
is convex in t.
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
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
λ
<
D
k
and every
ϵ
>
0
there exists a
δ
=
δ
(
k
,
λ
,
ϵ
)
>
0
such that, if
m
is the midpoint 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
>
0
.
An
n
-dimensional
CAT
(
k
)
space equipped with the
n
-dimensional Hausdorff measure satisfies the
CD
[
n
,
(
n
−
1
)
k
]
condition in the sense of Lott-Villani-Sturm.
