On complete Riemannian manifolds, there is a characterization of the time $t_0$ when a geodesic $c$ stops being minimizing: either $c(t_0)$ is conjugate to $c(0)$ along $c$, or there exists a geodesic $\gamma$ from $c(0)$ to $c(t_0)$ with $\gamma \neq c$ and $L(\gamma) = L(c)$. (See Cheeger--Ebin, Comparison Theorems in Riemannian Geometry, Lemma 5.2.)

Does anyone know if there is a similar (at least partial) characterization for incomplete Riemannian manifolds? Any references would be much appreciated.

Thanks in advance.