Let $M$ be a compact two-dimensional manifold. I'm interested in geodesic vector fields on $M$: smooth vector fields $v$ that have the property that every integral curve of the vector field is (locally) a geodesic. In other words $$\nabla_v v = \phi v$$ where $\phi$ is a function $M\to\mathbb{R}$.
What is known about the space of such vector fields? Is there a way to characterize or parameterize them?
For example, I imagine one possible construction is to pick a closed subset $S$ of $M$, construct the signed distance function $d_S$ to $S$, and set $v = \nabla d_S$. This vector field is undefined at the cut locus of $S$, but presumably that can be fixed by adjusting the magnitude of $v$ so that it vanishes near the cut locus.