Intrinsic Metric - Examples

Examples

• Euclidean space Rn with the ordinary Euclidean metric is a path metric space. Rn - {0} is as well.
• The unit circle S1 with the metric inherited from the Euclidean metric of R2 (the chordal metric) is not a path metric space. The induced intrinsic metric on S1 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. (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 (Khamsi & Kirk 2001, Theorem 2.16), a result of Karl Menger. The converse does not hold in general, however: there are length metric spaces which are not convex.