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 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 then it may be 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 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. 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
The unit circle with the metric inherited from the Euclidean metric of 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 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. Analogously, other manifolds in which a length is defined included Finsler manifolds and sub-Riemannian manifolds.
