Intrinsic metric - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

study of metric spaces, one can consider the arclength of paths in the space. If two points are a given distance from each other, it is natural to expect that one should be able to get from one point to another 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

Infimum

In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...

of the length of all paths from one point to the other. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space.

Definitions

Let

be a metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

. We define a new metric

, known as the induced intrinsic metric, as follows:

is the infimum

Infimum

of the lengths of all paths from

.

Here, a path from

is a continuous map

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

. If

for all points

and

, 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

there exists

such that

and

are both smaller than

Examples

Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

Rⁿ with the ordinary Euclidean metric is a path metric space. Rⁿ - {0} is as well.
The unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

S¹ with the metric inherited from the Euclidean metric of R² (the chordal metric) is not a path metric space. The induced intrinsic metric on S¹ measures distances as angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s, and the resulting length metric space is called the Riemannian circle
Riemannian circle
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle equipped with its great-circle distance...

. In two dimensions, the chordal metric on the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

is not intrinsic, and the induced intrinsic metric is given by the great-circle distance
Great-circle distance
The great-circle distance or orthodromic distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere . Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a...

.
Every Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

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
Finsler manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold together with the structure of an intrinsic quasimetric space in which the length of any rectifiable curve is given by the length functional...

s and sub-Riemannian manifold
Sub-Riemannian manifold
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces....

s.
Any complete and convex metric space
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....

is a length metric space , a result of Karl Menger
Karl Menger
Karl Menger was a mathematician. He was the son of the famous economist Carl Menger. He is credited with Menger's theorem. He worked on mathematics of algebras, algebra of geometries, curve and dimension theory, etc...

. The converse does not hold in general, however: there are length metric spaces which are not convex.

Properties

In general, we have d ≤ d_l and the topology
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

defined by d_l is therefore always finer than or equal to the one defined by d.
The space (M, d_l) is always a path metric space (with the caveat, as mentioned above, that d_l can be infinite).
The metric of a length space has approximate midpoints. Conversely, every complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

metric space with approximate midpoints is a length space.
The Hopf–Rinow theorem
Hopf–Rinow theorem
In mathematics, the 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 .-Statement of the theorem:...

states that if a length space is complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

and locally compact then any two points in can be connected by a minimizing geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

and all bounded closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

s in are compact.

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