If two places are at a distance one mile from each other, it is reasonable to expect that it should be possible to construct a road of length one mile between them. Except, of course, if there's a lake in the way. In mathematics, the general notion of measuring distances is captured with abstract metric spaces. If in such a metric space the distance between any two points can be realized with a "road" of the same length, we call the metric space a length space and the metric intrinsic.

Definitions

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Suppose (M, d) is a metric space. We define a new metric d_l on M, known as the induced intrinsic metric, as follows: d_l(x,y) is the infimum of the lengths of all paths from x to y. Here, a path from x to y is a continuous map γ : ^* → M with γ(0) = x and γ(1) = y. The length of such a path is defined as explained for rectifiable curves. We set d_l(x, y) = ∞ if there is no path of finite length from x to y.

If d(x,y) = d_l(x,y) for all points x and y in M, we say (M, d) is a length space or a path metric space and the metric d is intrinsic.

We say that the metric d has approximate midpoints if for any ε>0 and any pair of points x, y in M there exists c in M such that d(x,c) and d(c,y) are both smaller than d(x,y)/2 + ε.

Examples

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• Euclidean space Rⁿ with the ordinary Euclidean metric is a path metric space. Rⁿ - {0} is as well.
• If we consider the unit circle S¹ as a metric space with the metric inherited from the Euclidean metric of R², then it is not a path metric space. The induced intrinsic metric on S¹ measures distances as angles in radians.
• 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 to define the length of such curves)
• Finsler manifolds.
• sub-Riemannian manifolds.

Properties

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In general, we have d ≤ d_l and the topology defined by d_l is therefore always coarser than or equal to the one defined by d.

The space (M, d_l) is always a path metric space (with the caveat, as mentioned above, that d_l can be infinite).

The metric of a length space has approximate midpoints. Conversely, every complete metric space with approximate midpoints is a length space.

Hopf-Rinow theorem

The Hopf-Rinow theorem states that if a length space $(M,d)$ is complete and locally compact then any two points in $M$ can be connected by a minimizing geodesic and all bounded closed sets in $M$ are compact.

References

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• Mikhail Gromov. Metric structures for Riemannian and non-Riemannian spaces. Progress in Math., 152, Birkhäuser (1999). ISBN 0817638989

Metric geometry