Motivated by the "The obvious Fact" part of this answer,, we ask the following question:

First we recall a definition, which is used in the above link:

Definition: A non vanishing vector field $V$ on a manifold $M$ is a geodesible vector field if there is a Riemannian metric $g$ on $M$ such that all trajectories of $V$ are (unparametrized) geodesics with respect to $g$.

Assume that $U$ is an open subset of $\mathbb{C}^n$ and $V:U \to \mathbb{C}^n$ is a non vanishing holomorphic function. This holomorphic function $V$ defines a natural vector field on $U\subseteq \mathbb{R}^{2n}$. (Here we identify $\mathbb{C}^{n}$ with $\mathbb{R}^{2n}$). Is $V$ necessarily a geodesible flow?

Remark: The concept "geodesible flow" can be used to count the number of limit cycles of a planar vector field. In this regard please see the following posts:

Flat Riemannian metrics adapted to quadratic vector fields with center

Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)


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