A unit disk with the Euclidean metric should be complete as a metric space.

But it is not geodesically complete I guess. Since every line will reach the boundary and can not be defined for all $t$.

But the Hopf–Rinow theorem says these two concepts are equivalent.

What is wrong in the above arguments...


  • $\begingroup$ Why should the disk be complete? That's a weird statement, given that it is not! $\endgroup$ – Mariano Suárez-Álvarez Apr 6 '16 at 5:19
  • $\begingroup$ I mean closed disk... $\endgroup$ – tomography Apr 6 '16 at 5:20
  • $\begingroup$ Then it is not a manifold, so the theorem does not say anything about it! $\endgroup$ – Mariano Suárez-Álvarez Apr 6 '16 at 5:21

If you're talking about the open unit disk, it's not complete (sequences approaching a point on the boundary are Cauchy but not convergent).

If you're talking about the closed unit disk, it's not a manifold but a manifold-with-boundary. Hopf-Rinow applies only to manifolds without boundary.

  • 1
    $\begingroup$ Whops... Thanks a lot. $\endgroup$ – tomography Apr 6 '16 at 5:23

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