Quasi-isometries

Quasi-isometries[edit]

A quasi-isometry is a map that preserves the "large-scale structure" of a metric space. Quasi-isometries need not be continuous. For example, $\mathbb {R} ^{2}$ and its subspace $\mathbb {Z} ^{2}$ are quasi-isometric, even though one is connected and the other is discrete. The equivalence relation of quasi-isometry is important in geometric group theory: the Švarc–Milnor lemma states that all spaces on which a group acts geometrically are quasi-isometric.^[15]

Formally, the map $f\,\colon M_{1}\to M_{2}$ is a quasi-isometric embedding if there exist constants $A \geq 1$ and $B \geq 0$ such that

{\frac {1}{A}}d_{2}(f(x),f(y))-B\leq d_{1}(x,y)\leq Ad_{2}(f(x),f(y))+B\quad {\text{ for all }}\quad x,y\in M_{1}.

It is a quasi-isometry if in addition it is quasi-surjective, i.e. there is a constant

C \geq 0

such that every point in

M_{2}

is at distance at most

C

from some point in the image

f(M_{1})

Notions of metric space equivalence[edit]

Given two metric spaces $(M_{1},d_{1})$ and $(M_{2},d_{2})$ :

They are called homeomorphic (topologically isomorphic) if there is a homeomorphism between them (i.e., a continuous bijection with a continuous inverse). If $M_{1}=M_{2}$ and the identity map is a homeomorphism, then $d_{1}$ and $d_{2}$ are said to be topologically equivalent.
They are called uniformic (uniformly isomorphic) if there is a uniform isomorphism between them (i.e., a uniformly continuous bijection with a uniformly continuous inverse).
They are called bilipschitz homeomorphic if there is a bilipschitz bijection between them (i.e., a Lipschitz bijection with a Lipschitz inverse).
They are called isometric if there is a (bijective) isometry between them. In this case, the two metric spaces are essentially identical.
They are called quasi-isometric if there is a quasi-isometry between them.

Metric spaces with additional structure[edit]

Normed vector spaces[edit]

A normed vector space is a vector space equipped with a norm, which is a function that measures the length of vectors. The norm of a vector $v$ is typically denoted by $\lVert v\rVert$ . Any normed vector space can be equipped with a metric in which the distance between two vectors $x$ and $y$ is given by

d(x,y)=\lVert x-y\rVert .

The metric

d

is said to be induced by the norm

\lVert {\cdot }\rVert

. Conversely,^[16] if a metric

d

on a vector space

X

translation invariant: $d(x,y)=d(x+a,y+a)$ for every $x$ , $y$ , and $a$ in $X$ ; and
absolutely homogeneous: $d(\alpha x,\alpha y)=|\alpha |d(x,y)$ for every $x$ and $y$ in $X$ and real number $α$ ;

then it is the metric induced by the norm

\lVert x\rVert =d(x,0).

A similar relationship holds between seminorms and pseudometrics.

Among examples of metrics induced by a norm are the metrics $d 1$ , $d 2$ , and $d \infty$ on $\mathbb {R} ^{2}$ , which are induced by the Manhattan norm, the Euclidean norm, and the maximum norm, respectively. More generally, the Kuratowski embedding allows one to see any metric space as a subspace of a normed vector space.

Infinite-dimensional normed vector spaces, particularly spaces of functions, are studied in functional analysis. Completeness is particularly important in this context: a complete normed vector space is known as a Banach space. An unusual property of normed vector spaces is that linear transformations between them are continuous if and only if they are Lipschitz. Such transformations are known as bounded operators.

Length spaces[edit]

One possible approximation for the arc length of a curve. Notice that the approximation is never longer than the arc length, justifying the definition of arc length as a supremum.

A curve in a metric space $(M, d)$ is a continuous function $\gamma :[0,T]\to M$ . The length of $γ$ is measured by

L(\gamma )=\sup _{0=x_{0}<x_{1}<\cdots <x_{n}=T}\left\{\sum _{k=1}^{n}d(\gamma (x_{k-1}),\gamma (x_{k}))\right\}.

In general, this supremum may be infinite; a curve of finite length is called rectifiable.^[17] Suppose that the length of the curve

γ

is equal to the distance between its endpoints—that is, it's the shortest possible path between its endpoints. After reparametrization by arc length,

γ

becomes a geodesic: a curve which is a distance-preserving function.^[15] A geodesic is a shortest possible path between any two of its points.^[d]

A geodesic metric space is a metric space which admits a geodesic between any two of its points. The spaces $(\mathbb {R} ^{2},d_{1})$ and $(\mathbb {R} ^{2},d_{2})$ are both geodesic metric spaces. In $(\mathbb {R} ^{2},d_{2})$ , geodesics are unique, but in $(\mathbb {R} ^{2},d_{1})$ , there are often infinitely many geodesics between two points, as shown in the figure at the top of the article.

The space $M$ is a length space (or the metric $d$ is intrinsic) if the distance between any two points $x$ and $y$ is the infimum of lengths of paths between them. Unlike in a geodesic metric space, the infimum does not have to be attained. An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points $(1, 0)$ and $(-1, 0)$ can be joined by paths of length arbitrarily close to 2, but not by a path of length 2. An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between two points through the center of the earth is shorter than any path along the surface.

Given any metric space $(M, d)$ , one can define a new, intrinsic distance function $d intrinsic$ on $M$ by setting the distance between points $x$ and $y$ to be infimum of the $d$ -lengths of paths between them. For instance, if $d$ is the straight-line distance on the sphere, then $d intrinsic$ is the great-circle distance. However, in some cases $d intrinsic$ may have infinite values. For example, if $M$ is the Koch snowflake with the subspace metric $d$ induced from $\mathbb {R} ^{2}$ , then the resulting intrinsic distance is infinite for any pair of distinct points.

Riemannian manifolds[edit]

A Riemannian manifold is a space equipped with a Riemannian metric tensor, which determines lengths of tangent vectors at every point. This can be thought of defining a notion of distance infinitesimally. In particular, a differentiable path $\gamma :[0,T]\to M$ in a Riemannian manifold $M$ has length defined as the integral of the length of the tangent vector to the path:

L(\gamma )=\int _{0}^{T}|{\dot {\gamma }}(t)|dt.

On a connected Riemannian manifold, one then defines the distance between two points as the infimum of lengths of smooth paths between them. This construction generalizes to other kinds of infinitesimal metrics on manifolds, such as sub-Riemannian and Finsler metrics.

The Riemannian metric is uniquely determined by the distance function; this means that in principle, all information about a Riemannian manifold can be recovered from its distance function. One direction in metric geometry is finding purely metric ("synthetic") formulations of properties of Riemannian manifolds. For example, a Riemannian manifold is a $CAT(k)$ space (a synthetic condition which depends purely on the metric) if and only if its sectional curvature is bounded above by $k$ .^[20] Thus $CAT(k)$ spaces generalize upper curvature bounds to general metric spaces.

Metric measure spaces[edit]

Real analysis makes use of both the metric on $\mathbb {R} ^{n}$ and the Lebesgue measure. Therefore, generalizations of many ideas from analysis naturally reside in metric measure spaces: spaces that have both a measure and a metric which are compatible with each other. Formally, a metric measure space is a metric space equipped with a Borel regular measure such that every ball has positive measure.^[21] For example Euclidean spaces of dimension $n$ , and more generally $n$ -dimensional Riemannian manifolds, naturally have the structure of a metric measure space, equipped with the Lebesgue measure. Certain fractal metric spaces such as the Sierpiński gasket can be equipped with the α-dimensional Hausdorff measure where α is the Hausdorff dimension. In general, however, a metric space may not have an "obvious" choice of measure.

One application of metric measure spaces is generalizing the notion of Ricci curvature beyond Riemannian manifolds. Just as $CAT(k)$ and Alexandrov spaces generalize scalar curvature bounds, RCD spaces are a class of metric measure spaces which generalize lower bounds on Ricci curvature.^[22]

Further examples and applications[edit]

Graphs and finite metric spaces[edit]

A metric space is discrete if its induced topology is the discrete topology. Although many concepts, such as completeness and compactness, are not interesting for such spaces, they are nevertheless an object of study in several branches of mathematics. In particular, finite metric spaces (those having a finite number of points) are studied in combinatorics and theoretical computer science.^[23] Embeddings in other metric spaces are particularly well-studied. For example, not every finite metric space can be isometrically embedded in a Euclidean space or in Hilbert space. On the other hand, in the worst case the required distortion (bilipschitz constant) is only logarithmic in the number of points.^[24]^[25]

For any undirected connected graph $G$ , the set $V$ of vertices of $G$ can be turned into a metric space by defining the distance between vertices $x$ and $y$ to be the length of the shortest edge path connecting them. This is also called shortest-path distance or geodesic distance. In geometric group theory this construction is applied to the Cayley graph of a (typically infinite) finitely-generated group, yielding the word metric. Up to a bilipschitz homeomorphism, the word metric depends only on the group and not on the chosen finite generating set.^[15]

Distances between mathematical objects[edit]

In modern mathematics, one often studies spaces whose points are themselves mathematical objects. A distance function on such a space generally aims to measure the dissimilarity between two objects. Here are some examples:

Functions to a metric space. If $X$ is any set and $M$ is a metric space, then the set of all bounded functions $f\colon X\to M$ (i.e. those functions whose image is a bounded subset of $�$ ) can be turned into a metric space by defining the distance between two bounded functions $f$ and $g$ to be $d(f,g)=\sup _{x\in X}d(f(x),g(x)).$ This metric is called the uniform metric or supremum metric.^[26] If $M$ is complete, then this function space is complete as well; moreover, if $X$ is also a topological space, then the subspace consisting of all bounded continuous functions from $X$ to $M$ is also complete. When $X$ is a subspace of $\mathbb {R} ^{n}$ , this function space is known as a classical Wiener space.
String metrics and edit distances. There are many ways of measuring distances between strings of characters, which may represent sentences in computational linguistics or code words in coding theory. Edit distances attempt to measure the number of changes necessary to get from one string to another. For example, the Hamming distance measures the minimal number of substitutions needed, while the Levenshtein distance measures the minimal number of deletions, insertions, and substitutions; both of these can be thought of as distances in an appropriate graph.
Graph edit distance is a measure of dissimilarity between two graphs, defined as the minimal number of graph edit operations required to transform one graph into another.
Wasserstein metrics measure the distance between two measures on the same metric space. The Wasserstein distance between two measures is, roughly speaking, the cost of transporting one to the other.
The set of all $m$ by $n$ matrices over some field is a metric space with respect to the rank distance $d(A,B)=\mathrm {rank} (B-A)$ .
The Helly metric in game theory measures the difference between strategies in a game

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