# Complete manifold and the Hopf–Rinow theorem

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...

Thanks,

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