Intrinsic metric

  (Redirected from Length metric space)

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 (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 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 (a geodesic) 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.

DefinitionsEdit

Let   be a metric space, i.e.,   is a collection of points (such as all of the points in the plane, or all points on the circle) 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

 .

ExamplesEdit

  • Euclidean space   with the ordinary Euclidean metric is a path metric space.   is as well.
  • The unit circle   with the metric inherited from the Euclidean metric of   (the chordal metric) 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. (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.

PropertiesEdit

  • In general, we have   and the topology defined by   is therefore always finer than or equal to the one defined by  .
  • The space   is always a path metric space (with the caveat, as mentioned above, that   can be infinite).
  • The metric of a length space has approximate midpoints. Conversely, every complete metric space with approximate midpoints is a length space.
  • The Hopf–Rinow theorem states that if a length space   is complete and locally compact then any two points in   can be connected by a minimizing geodesic and all bounded closed sets in   are compact.

ReferencesEdit

  • Gromov, Mikhail (1999), Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Math., 152, Birkhäuser, ISBN 0-8176-3898-9
  • Khamsi, Mohamed A.; Kirk, William A. (2001), An Introduction to Metric Spaces and Fixed Point Theory, Wiley-IEEE, ISBN 0-471-41825-0