If you will excuse me substituting a polyhedron for the Riemannian manifold
(imagine rounding the vertices),
this figure shows how the cut locus (red) from source $x$ is a tree.
(The green arcs are equidistant from $x$.)
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;![2x1x1Box][1]
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<sub>(Figure from [Discrete and Computational Geometry][2])</sub>


  [1]: https://i.sstatic.net/2RDoQ.png
  [2]: http://cs.smith.edu/~orourke/DCG/