Let $S$ be a patch of a smooth 2-manifold in $\mathbb{R}^3$, and pick two distinct points $a,\ b \in S$. Let $c$ be the set of points on $S$ equidistant to $a$ and $b$, where distance is defined by shortest paths on the surface. Call $c$ a *bisector*.

A bisector may be non-manifold at points (a torus with a long, thin rod attached has pairs of points with such bisectors). A bisector is, in a sense, a generalized kind of medial axis or Voronoi Diagram.

To illustrate, given a few points on $S$ we may construct a so called *geodesic Voronoi Diagram*. It's easy to see that it
is made of segments of bisectors:

The resulting figure looks analogous to a classical Voronoi Diagram. How far does the analogy extend?

Q.Suppose we have a connected 1-manifold segment of the bisector of $a$ and $b$. Is it a segment of a geodesic on $S$?.

Credit:

The illustration is from Franz Wolter's web article "Cut Locus and Medial Axis in the Euclidean Space and on Surfaces" (link: Wayback Machine)

I also shamelessly copied from Joseph O'Rourke's related MO post.