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The Space of Metric Spaces
*D. J. Kelleher1
1 Department
of Mathematics
University of Connecticut
UConn— SIGMA Seminar — Fall 2013
D. J. Kelleher
The Space of Metric Spaces
Intro Vamp
D. J. Kelleher
The Space of Metric Spaces
Basic metric space stuff
A metric space is a set that has some notion of distance
Definition (Metric Space and Distance Function)
A metric space (X, d), is a set X and a map d : X × X → R
which satisfies, for arbitrary x, y, z ∈ X,
1 (Symmetric) d(x, y) = d(y, x)
the distance from x to y is the same as the distance from y
to x.
2
(Positive Definite) d(x, y) ≥ 0 and d(x, y) = 0 if and only if
x = y.
The distance between different points is positive.
the distance between a point and itself is 0.
3
(Triangle inequality) d(x, z) ≤ d(x, y) + d(y, z).
It takes longer to get there if you make stops.
d is called a metric or distance function.
D. J. Kelleher
The Space of Metric Spaces
Examples of metric spaces
1
Euclidean space with Euclidean distance: (Rn , d) with
!1/2
n
X
d(x, y) =
|xi − yi |2
i=1
2
In fact, if V is a vector space with norm k−k , then
d(x, y) = kx − yk, for x, y ∈ V is a distance function.
3
Riemannian manifolds, Kahler manifolds, Finsler
manifolds, with intrinsic distances.
D. J. Kelleher
The Space of Metric Spaces
Metric spaces are topological spaces
Definition (Balls)
For x ∈ X, and r ∈ [0, ∞), define the set
B(x, r) := {y ∈ X | d(x, y) < r}
to be the ball of radius r centered at x.
That is that B(x, r) is the collection of all points in X which
are less than distance r away from x.
D. J. Kelleher
The Space of Metric Spaces
Metric spaces as topological spaces
The collection of balls of all radii and centers form a basis for a
topology on X is given by the basis:
B = {B(x, r) | x ∈ X, r ∈ (0, ∞)}
Many of the terms can be rephrased in a more natural way for
metric spaces...
1
Equivalently, U ⊂ X is open, if for all x ∈ U , there is
r > 0 such that B(x, r) ⊂ U .
2
Let (X, dX ), (Y, dY ) be metric spaces. f : X → Y is
continuous if, for a given x, ∀ε > 0, ∃δ > 0 such that
dX (x, y) < δ ⇒ dY (f (x), f (y)) < ε.
D. J. Kelleher
The Space of Metric Spaces
Convergence
In a metric space (X, d) a sequence {xi } converges to a point
x if limn→∞ d(x, xn ) = 0.
A sequence {xi } is Cauchy if limm,n→∞ d(xn , xm ) = 0.
(X, d) is Complete if every Cauchy sequence is convergent...
that is if {xi } is Cauchy, then there is x ∈ X with
limn→∞ xn = x.
Assumption: For the rest of the talk, I assume that all metric
spaces are complete.
D. J. Kelleher
The Space of Metric Spaces
Maps between metric spaces
Isometries: A bijection f : X → Y is an isometry if
dY (f (x), f (y)) = dX (x, y), that is f ∗ dY = dX .
Lipshitz maps:A map f : X → Y is Lipshitz, if there is
Cf ∈ [0, ∞) such that dY (f (x), f (y)) ≤ Cf dX (x, y). I’ll call Cf
the Lipshitz constant of f .
bi-Lipshitz maps: A Lipshitz map f : X → Y is bi-Lipshitz if
for some C ∈ (0, ∞),
dX (x, y)
≤ dY (f (x), f (y)) ≤ CdX (x, y).
C
Contraction: A Lipshitz map f : X → Y is a contraction if
Cf ∈ [0, 1).
D. J. Kelleher
The Space of Metric Spaces
Dan proves something!
Theorem (A fixed point theorem)
If (X, d) is a complete metric space, a contraction map
f : X → X has a unique fixed point.
Proof. Pick any x0 in X, define the sequence {xn } by
xn = f n (x0 ) (f n is f composed with itself n-times.) Assume
n ≤ m.
d(xn , xn+m ) ≤ Cfn d(x, xm )
≤ Cfn (d(x, x1 ) + d(x1 , x2 ) + · · · d(xm−1 , xm )
≤ Cfn d( x, x1 )(Cf + Cf2 + · · · + Cfm−1 )
≤
Cfn d(x, x1 )
1 − Cf
→ 0 with n.
So there is x = limn xn , and since f is continuous,
f (x) = limn xn+1 = x.
D. J. Kelleher
The Space of Metric Spaces
Distance between subsets of a metric space
We would like to have a notion of distance between two subsets
of a metric space....
What about if we take dist(U, V ) = inf x∈U,y∈V d(x, y)?
Ummm... that works well for distance between a point and a
set, but all it tells us is if some two points in a set are close. We
would like this distance to tell us if the sets are close to each
other.
So we shall define dist(x, U ) = inf y∈U d(x, y).
D. J. Kelleher
The Space of Metric Spaces
Distance between subsets of a metric space
We would like to have a notion of distance between two subsets
of a metric space....
What about if we take
dH (U, V ) =
sup
{dist(x, V ), dist(y, U )}?
x∈U,y∈V
1
This is clearly symmetric and positive.
2
Takes a little work, but it satisfies the triangle inequality.
We call this distance Hausdorff distance.
D. J. Kelleher
The Space of Metric Spaces
Squinting distance
Equivalently, dH (U, V ) is the smallest number such that for
all x ∈ U and ε > 0, there is y ∈ V with d(x, y) < dH (U, V ) + ε,
and visa versa.
U is Hausdorff close to V if they look the same when you squint.
D. J. Kelleher
The Space of Metric Spaces
Space of compact spaces
1
If dH (U, V ) = 0, then every point in U ⊆ V and V ⊆ U .
2
And dH (U, U ) = 0
Theorem
If H(x) := {K ⊂ X | K is compact}, then (H(X), dH ) is a
metric space.
D. J. Kelleher
The Space of Metric Spaces
Application: Iterated function systems
Definition
A set of contractions fi : X → X for i = 1, 2, . . . , N is called an
iterated function system or IFS for short.
Theorem
For {fi }N
i=1 is an IFS, then there is a unique compact K ⊂ X
such that
K = ∪N
i=1 fi (K).
Proof. It is easy to check that F : H(X) → J (X),
F (U ) =
N
[
fi (U )
i=1
is a contraction and thus has a unique fixed point.
D. J. Kelleher
The Space of Metric Spaces
You knew there would be fractals somewhere
Many fractals are constructed through iterated function
systems.
D. J. Kelleher
The Space of Metric Spaces
Gromov-Hausdorff distance
Given two arbitrary metric spaces (X, dX ), (Y, dY )
dGH (X, Y ) = inf dH
Z (φ(X), ψ(Y ))
Where the infimum is taken over all φ, ψ, Z such that
φ : X → Z, ψ : Y → Z are isometric embeddings.
That is, if GH is the set of all Compact metric spaces,
(GH, dGH ) is a metric space.
D. J. Kelleher
The Space of Metric Spaces
We say that a property is preserved by taking GH limits if
whenever a sequence of compact metric spaces {(Xi , di )} have
this property and (Xi , di ) → (X, d), then X also has this
property.
Example Diameter: Take Diam(X) = supx,y∈X d(x, y),
If Diam(Xi ) ≤ D, then Diam(X) ≤ D.
In fact if XD = {X ∈ GH | Diam(X) ≤ D} itself has diameter
2D.
D. J. Kelleher
The Space of Metric Spaces
Length of Curves
A Curve in X is a continuous function γ : [a, b] → X. The
Length of a curve γ is defined
(N
)
X
L(γ) = sup
d(γ(ti−1 ), γ(ti )) | a = t0 ≤ t1 ≤ · · · ≤ tN = b .
i=1
1
d(γ(a), γ(b)) ≤ L(γ).
2
L is invariant under reparametrization, that is if
φ : [a, b] → R is a monotone increasing function
L(γ) = L(γ ◦ φ)
Because of 2 above, we can assume wolog, that γ is a Unit
speed curve — for any c, d ∈ [a, b], c < d, L(γ|[c,d] ) = d − c.
D. J. Kelleher
The Space of Metric Spaces
If things are smooth...
Exercise. Show, for X = Rn
Z b dγ dt.
L(γ) =
dt a
(If you are feeling feisty, try it for a Riemannian manifold)
Hint:
D. J. Kelleher
The Space of Metric Spaces
Length spaces
A metric space (X, d) is a Length Space if
d(x, y) = inf {L(γ) | γ(a) = x, γ(b) = y} .
In particular, X is path connected.
A curve γ in a length space is called Length minimizing or a
Geodesic of L(γ) = d(γ(a), γ(b)).
A Lengths space X is called Strictly intrinsic or Geodesic if
for all pairs of points can be connected with a geodesic,
x, y ∈ X, there is a length minimizing curve γ with γ(a) = x,
γ(b) = y.
D. J. Kelleher
The Space of Metric Spaces
Length spaces and GH limits
A metric space (X, d) is a Length Space if
d(x, y) = inf {L(γ) | γ(a) = x, γ(b) = y} .
In particular, X is path connected.
The property of being a length space is preserved by taking
GH limits. That is if (Xi , di ) are length spaces, and
(Xi , di ) → (X, d) in the GH metric, then (X, d) is a length
space.
D. J. Kelleher
The Space of Metric Spaces
Examples
1
2
In Rn the shortest path between two points is a line.
(Unique)
Taxi cab metric In R2 , with the `1 -norm
k(x, y)k = |x| + |y|,
Very not unique
D. J. Kelleher
The Space of Metric Spaces
examples
Length minimizers on a sphere are arcs of great circles.
D. J. Kelleher
The Space of Metric Spaces
Comparison Triangles
A triangle in a geodesic space 4xyz is three points x, y, z ∈ X
and length minimizing curves between them, which we shall
refer to the ranges of these curves as [xy], [yz], [zx].
xe
x
ze
z
ye
y
Given a triangle 4xyz in X, a Euclidean Comparison
triangle 4e xyz is a triangle with verteces xe , y e , z e in R2 such
that |xe − y e | = d(x, y), etc.
D. J. Kelleher
The Space of Metric Spaces
Curvature by triangle comparison
We say that a geodesic space (X, d) is non-negatively curved
(resp. non-positive) if for every point p ∈ X there is a
neighborhood of p such that for all triangles 4xyz in this
neighborhood the following condition holds:
x
x
wew
w
we
z
y
z
y
if w ∈ [xy], and we is the corresponding point in the Euclidean
comparison triangle, then d(z, w) ≥ |z e − we | (resp
d(z, w) ≤ |ze − we |).
D. J. Kelleher
The Space of Metric Spaces
Eeeek! A spider!
Walsh spider Three copies of [0, ∞) glued at 0
Triangles are either Euclidean (if they are contained in one or
two copy of [0, ∞)) or thin. So it has non-positive curvature.
D. J. Kelleher
The Space of Metric Spaces
Taxi cab metric
Triangles aren’t unique, and if the verteces aren’t contained in a
single horizontal/vertical line, there are fat and thin triangles.
D. J. Kelleher
The Space of Metric Spaces
Relation to smooth spaces
Theorem (Toponogrov’s theorem)
Riemannian manifolds have non-negative (resp.
non-positive) curvature if and only if the sectional curvature
κ is bounded below (resp. above) by 0
We can generalize the notion to having curvature bounded
above or below by a constant by choosing comparison triangles
from spaces with constant sectional curvature of that constant.
D. J. Kelleher
The Space of Metric Spaces
Other model spaces
Positively curved: Spheres Sn : triangles are fatter than their
comparisons in Rn . Scalar curvature of 1/R2 , where R is the
raidius of the sphere.
D. J. Kelleher
The Space of Metric Spaces
Other model spaces
Negative Curvature: Hyperbolic spaces Hn : Triangles are
thinner than their comparisons.
D. J. Kelleher
The Space of Metric Spaces
Gromov’s compactness theorem
Theorem (Gromov’s compactness theorem)
If M(n, k, D) is the space of spaces with curvature bounded
below by k, with diameter less than D and Hausdorff dimension
less than n, then M(n, k, D) is a compact space with respect to
dGH .
Note If k > 0 then the upper bound on diameter is superfluous.
1
The subset of of spaces with diameter bounded by D is
bounded.
2
The subset of length spaces is closed.
3
The subset of curvature bounded below by k are closed.
4
The subset of dimension bounded above by n is closed.
D. J. Kelleher
The Space of Metric Spaces
Gromov’s pre-compactness theorem
Theorem (Gromov’s pre-compactness theorem)
The space of n-dimensional Riemannian manifolds of diameter
no greater than D and sectional curvature K ≥ k is pre-compact
with respect to dGH .
D. J. Kelleher
The Space of Metric Spaces
Thank you!!!
D. J. Kelleher
The Space of Metric Spaces