Here, I try to give a exemple for each of the three curves with a convex polyhedron where I just draw the net. The geodesic are the dots lines (which are staight on the net). But I don't know any theorem that gives a general answer for your question. My only gess is that if you don't have a trivial estimate that says that on a subset $B$,  $\int_B K \geq 4\pi $, then we should be able to construct a exemple. 
[![enter image description here][1]][1]
[![enter image description here][2]][2]
[![enter image description here][3]][3]


  [1]: https://i.sstatic.net/LNoH4.png
  [2]: https://i.sstatic.net/otkxy.png
  [3]: https://i.sstatic.net/3prhR.png