Intrinsic Metric - Properties

Properties

  • In general, we have d ≤ dl and the topology defined by dl is therefore always finer than or equal to the one defined by d.
  • The space (M, dl) is always a path metric space (with the caveat, as mentioned above, that dl 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.

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