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 /> ![2x1x1Box][1] <br /> <sub>(Figure from [Discrete and Computational Geometry][2])</sub> <br /> Concerning, "What are the edges?": They are geodesics, such that there are two distinct shortest paths from $x$ to every interior point of an edge of the cut locus. Points of the cut locus of degree $k$ have $k$ distinct shortest paths from $x$. [1]: https://i.sstatic.net/2RDoQ.png [2]: http://cs.smith.edu/~jorourke/DCG/