Intrinsic metric
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If two objects are at a distance one mile from each other, it should be possible to construct a road of length one mile between them. That seems to be a reasonable expectation; but in mathematics it fails to be true for a general metric space. For example, taking the Earth's surface a straight road between north and south pole through the center of the Earth will not be considered as "possible" by most people. Metrics which satisfy the above property are called intrinsic. What follows next is the formal mathematical way to describe it.
Definition and discussion
A metric space <math>(M,d)<math> is called length space or path metric space or equivalently the metric <math>d<math> is called intrinsic if the distance between any pair of points in <math>M<math> is equal to the infimum of lengths of curves connecting these points. Equivalently <math>d<math> is intrinsic if for any <math>\epsilon>0<math> and any pair of points <math>x,y\in M<math> there is <math>c\in M<math> such that <math>2d(x,c)<math> and <math>2d(c,y)<math> are smaller than <math>d(x,y)+\epsilon<math>.
The Hopf-Rinow theorem states that if a length space <math>(M,d)<math> is complete and locally compact then any two points in <math>M<math> can be connected by minimizing geodesic and any bounded closed sets in <math>M<math> are compact. It is due to Heinz Hopf and his student Willi Rinow.
Given any metric <math>d<math>, one can define the induced intrinsic metric <math>\hat d<math> by saying <math>\hat d(x,y)<math> is to be the infimum of lengths of paths connecting <math>x<math> and <math>y<math> (or <math>\infty<math> if there is no contractible path connecting <math>x<math> and <math>y<math>). Clearly
- <math>\hat d\ge d<math>.
In general the topology defined by <math>\hat d<math> can be coarser than the one defined by <math>d<math>.