Intrinsic metric

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  be a metric space, 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 of the lengths of all paths from  to .

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

The mapping  is idempotent, i.e.

 

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  with the ordinary Euclidean metric is a path metric space.  is as well.
 The unit circle  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 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.
 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 manifolds and sub-Riemannian manifolds.
 Any complete and convex metric space is a length metric space , a result of Karl Menger.  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 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 metric space with approximate midpoints is a length space.
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.

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.

 

Metric geometry