information:extraction
Length spaces

A curve in a metric space (M, d) is a continuous function γ: → M {\displaystyle \gamma :\to M}. The length of γ is measured by L (γ) = sup 0 = x 0 < x 1 < ⋯ < x n = T { ∑ k = 1 n d (γ (x k − 1), γ (x k)) }. {\displaystyle L(\gamma)=\sup _{0=x_{0}<x_{1}<\cdots <x_{n}=T}\left\{\sum _{k=1}^{n}d(\gamma (x_{k-1}),\gamma (x_{k}))\right\}.} In general, this supremum may be infinite; a curve of finite length is called rectifiable. Suppose that the length of the curve γ is equal to the distance between its endpoints—that is, it is the shortest possible path between its endpoints. After reparametrization by arc length, γ becomes a geodesic: a curve which is a distance-preserving function. A geodesic is a shortest possible path between any two of its points.

A geodesic metric space is a metric space which admits a geodesic between any two of its points. The spaces (R 2, d 1) {\displaystyle (\mathbb {R} ^{2},d_{1})} and (R 2, d 2) {\displaystyle (\mathbb {R} ^{2},d_{2})} are both geodesic metric spaces. In (R 2, d 2) {\displaystyle (\mathbb {R} ^{2},d_{2})}, geodesics are unique, but in (R 2, d 1) {\displaystyle (\mathbb {R} ^{2},d_{1})}, there are often infinitely many geodesics between two points, as shown in the figure at the top of the article.

The space M is a length space (or the metric d is intrinsic) if the distance between any two points x and y is the infimum of lengths of paths between them. Unlike in a geodesic metric space, the infimum does not have to be attained. An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points (1, 0) and (-1, 0) can be joined by paths of length arbitrarily close to 2, but not by a path of length 2. An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between two points through the center of the Earth is shorter than any path along the surface.

Given any metric space (M, d), one can define a new, intrinsic distance function d on M by setting the distance between points x and y to be infimum of the d -lengths of paths between them. For instance, if d is the straight-line distance on the sphere, then d is the great-circle distance. However, in some cases d may have infinite values. For example, if M is the Koch snowflake with the subspace metric d induced from R 2 {\displaystyle \mathbb {R} ^{2}}, then the resulting intrinsic distance is infinite for any pair of distinct points.