Intrinsic metric explained

In the mathematical study of metric spaces, one can consider the arclength of paths 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 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 removed is not geodesic, but is still a length metric space.

Definitions

Let

(M,d)

be a metric space, i.e.,

M

is a collection of points (such as all of the points in the plane, or all points on the circle) and

d(x,y)

is a function that provides us with the distance between points

x,y\inM

. We define a new metric

dI

on

M

, known as the induced intrinsic metric, as follows:

dI(x,y)

is the infimum of the lengths of all paths from

x

to

y

.

Here, a path from

\gamma\colon[0,1]M

with

\gamma(0)=x

and

\gamma(1)=y

. The length of such a path is defined as explained for rectifiable curves. We set

dI(x,y)=infty

if there is no path of finite length from

x

to

y

(this is consistent with the infimum definition since the infimum of the empty set within the closed interval [0,+∞] is +∞).

The mapping d\mapsto d_\text is idempotent, i.e.

(dI)I=dI.

If

dI(x,y)=d(x,y)

for all points

x

and

y

in

M

, we say that

(M,d)

is a length space or a path metric space and the metric

d

is intrinsic.

We say that the metric

d

has approximate midpoints if for any

\varepsilon>0

and any pair of points

x

and

y

in

M

there exists

c

in

M

such that

d(x,c)

and

d(c,y)

are both smaller than

{d(x,y)\over2}+\varepsilon.

Examples

\Rn

with the ordinary Euclidean metric is a path metric space.

\Rn\smallsetminus\{0\}

is as well.

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 angles in radians, and the resulting length metric space is called the Riemannian circle. In two dimensions, the chordal metric on the sphere is not intrinsic, and the induced intrinsic metric is given by the great-circle distance.

Properties

d\ledI

and the topology defined by

dI

is therefore always finer than or equal to the one defined by

d

.

(M,dI)

is always a path metric space (with the caveat, as mentioned above, that

dI

can be infinite).

(M,d)

is complete and locally compact then any two points in

M

can be connected by a minimizing geodesic and all bounded closed sets in

M

are compact.

References