Intrinsic metric

In themathematicalstudy ofmetric spaces,one can consider thearclengthofpathsin 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 theintrinsic metricis defined as theinfimumof the lengths of all paths from the first point to the second. A metric space is alength metric spaceif 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 (ageodesic) then it is called ageodesic metric spaceorgeodesic space.For instance, theEuclidean planeis a geodesic space, withline segmentsas its geodesics. The Euclidean plane with theoriginremoved is not geodesic, but is still a length metric space.

Definitions

Let $(M,d)$ be ametric 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 thedistancebetween points $x,y\in M$ .We define a new metric $d_{\text{I}}$ on $M$ ,known as theinduced intrinsic metric,as follows: $d_{\text{I}}(x,y)$ is theinfimumof the lengths of all paths from $x$ to $y$ .

Here, apathfrom $x$ to $y$ is acontinuous map

\gamma \colon [0,1]\rightarrow M

with $\gamma (0)=x$ and $\gamma (1)=y$ .Thelengthof such a path is defined as explained forrectifiable curves.We set $d_{\text{I}}(x,y)=\infty$ if there is no path of finite length from $x$ to $y$ (this is consistent with the infimum definition since the infimum of theempty setwithin the closed interval [0,+∞] is +∞).

The mapping ${\textstyle d\mapsto d_{\text{I}}}$ isidempotent,i.e.

(d_{\text{I}})_{\text{I}}=d_{\text{I}}.

If

d_{\text{I}}(x,y)=d(x,y)

for all points $x$ and $y$ in $M$ ,we say that $(M,d)$ is alength spaceor apath metric spaceand the metric $d$ isintrinsic.

We say that the metric $d$ hasapproximate midpointsif 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) \over 2}+\varepsilon.

Examples

Euclidean space $\mathbb {R} ^{n}$ with the ordinary Euclidean metric is a path metric space. $\mathbb {R} ^{n}\smallsetminus \{0\}$ is as well.
Theunit circle $S^{1}$ with the metric inherited from the Euclidean metric of $\mathbb {R} ^{2}$ (thechordal metric) is not a path metric space. The induced intrinsic metric on $S^{1}$ measures distances asanglesinradians,and the resulting length metric space is called theRiemannian circle.In two dimensions, the chordal metric on thesphereis not intrinsic, and the induced intrinsic metric is given by thegreat-circle distance.
Every connectedRiemannian manifoldcan 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 includedFinsler manifoldsandsub-Riemannian manifolds.
Anycompleteandconvex metric spaceis a length metric space (Khamsi & Kirk 2001,Theorem 2.16), a result ofKarl Menger.However, the converse does not hold, i.e. there exist length metric spaces that are not convex.

Properties

In general, we have $d\leq d_{\text{I}}$ and thetopologydefined by $d_{\text{I}}$ is therefore alwaysfinerthan or equal to the one defined by $d$ .
The space $(M,d_{\text{I}})$ is always a path metric space (with the caveat, as mentioned above, that $d_{\text{I}}$ can be infinite).
The metric of a length space has approximate midpoints. Conversely, everycompletemetric space with approximate midpoints is a length space.
TheHopf–Rinow theoremstates that if a length space $(M,d)$ is complete andlocally compactthen any two points in $M$ can be connected by aminimizing geodesicand all boundedclosed setsin $M$ arecompact.