In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
study of
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s, one can consider the
arclength of
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
s in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
of the lengths of all paths from the first point to the second. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space.
If the space has the stronger property that there always exists a path that achieves the infimum of length (a
geodesic) then it is called a geodesic metric space or geodesic space. For instance, the
Euclidean plane is a geodesic space, with
line segments as its geodesics. The Euclidean plane with the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
removed is not geodesic, but is still a length metric space.
Definitions
Let
be a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
, i.e.,
is a collection of points (such as all of the points in the plane, or all points on the circle) and
is a function that provides us with the ''distance'' between points
. We define a new metric
on
, known as the induced intrinsic metric, as follows:
is the
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
of the lengths of all paths from
to
.
Here, a ''path'' from
to
is a
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
:
with
and
. The ''length'' of such a path is defined as explained for
rectifiable curves. We set
if there is no path of finite length from
to
. If
:
for all points
and
in
, we say that
is a length space or a path metric space and the metric
is intrinsic.
We say that the metric
has approximate midpoints if for any
and any pair of points
and
in
there exists
in
such that
and
are both smaller than
:
.
Examples
*
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
with the ordinary Euclidean metric is a path metric space.
is as well.
* The
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
with the metric inherited from the Euclidean metric of
(the chordal metric) is not a path metric space. The induced intrinsic metric on
measures distances as
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s, and the resulting length metric space is called the
Riemannian circle
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle with a characteristic length. It is the circle equipped with the ''intrinsic'' Riemannian metric of a compact one-dimensional manifold of total length 2, or ...
. In two dimensions, the chordal metric on the
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
is not intrinsic, and the induced intrinsic metric is given by the
great-circle distance.
* Every connected
Riemannian manifold can be turned into a path metric space by defining the distance of two points as the infimum of the lengths of continuously differentiable curves connecting the two points. (The Riemannian structure allows one to define the length of such curves.) Analogously, other manifolds in which a length is defined included
Finsler manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth c ...
s and
sub-Riemannian manifolds.
* Any
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and
convex metric space
In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints.
Formally, consider a metric space (''X'', ''d'') and let ''x ...
is a length metric space , a result of
Karl Menger
Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low- regularity ("rough") curves ...
. However, the converse does not hold, i.e. there exist length metric spaces that are not convex.
Properties
*In general, we have
and the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
defined by
is therefore always
finer than or equal to the one defined by
.
*The space
is always a path metric space (with the caveat, as mentioned above, that
can be infinite).
*The metric of a length space has approximate midpoints. Conversely, every
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
metric space with approximate midpoints is a length space.
*The
Hopf–Rinow theorem
Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931. Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem ...
states that if a length space
is complete and
locally compact then any two points in
can be connected by a
minimizing geodesic and all bounded
closed sets in
are
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
.
References
* Herbert Busemann, Selected Works, (Athanase Papadopoulos, ed.) Volume I, 908 p., Springer International Publishing, 2018.
* Herbert Busemann, Selected Works, (Athanase Papadopoulos, ed.) Volume II, 842 p., Springer International Publishing, 2018.
*
*{{citation
, authorlink1 = Mohamed Amine Khamsi
, last1 = Khamsi
, first1 = Mohamed A.
, authorlink2 = William Arthur Kirk
, last2 = Kirk
, first2 = William A.
, title = An Introduction to Metric Spaces and Fixed Point Theory
, publisher = Wiley-IEEE
, date = 2001
, pages =
, isbn = 0-471-41825-0
Metric geometry