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 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 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.