Intrinsic metric explained
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 is 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 (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
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
(this is consistent with the infimum definition since the infimum of the
empty set within the closed interval [0,+∞] is +∞).
The mapping is idempotent, i.e.
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
{d(x,y)\over2}+\varepsilon.
Examples
with the ordinary Euclidean metric is a path metric space.
is as well.
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 connected 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, a result of Karl Menger. However, the converse does not hold, i.e. there exist length metric spaces that are not convex.
Properties
and the
topology defined by
is therefore always
finer than or equal to the one defined by
.
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.
References
- Herbert Busemann, Selected Works, (Athanase Papadopoulos, ed.) Volume I, 908 p., Springer International Publishing, 2018.
- Herbert Busemann, Selected Works, (Athanase Papadopoulos, ed.) Volume II, 842 p., Springer International Publishing, 2018.