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.

Read more about this topic:  Intrinsic Metric

Famous quotes containing the word properties:

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)