Let $(M,g)$ be a smooth Riemannian manifold, let $C \subset M$ be a closed region with $C^2$ boundary, and let $x_0 \in \partial C$. Suppose there is an open neighbourhood $U_0 \subset M$ of $x_0$ and a $C^2$ signed-distance function
to $\partial C$ such that $C \cap U_0 = \{x \in U_0 : \rho(x) \le 0\}$, $\nabla \rho = \nu$ on $\partial C \cap U_0$, where $\nu$ is the outward unit normal, and
Then there exists an open neighbourhood $U \subset U_0$ of $x_0$ with the following property: if $p,q \in C \cap U$ and the minimizing geodesic segment from $p$ to $q$ contained in $U$ is unique, then the image of that geodesic segment is contained in $C \cap U$.